Applications of Definite Integral

7) APPLICATIONS OF DEFINITE INTEGRAL

Application of Definite Integrals1. The area bounded by the line y=mx,X-axis and the ordinates x=a, x=b,(a<b) is

a)m (b2−a2)

3sq .units b)

m (a2−b2)3

sq .units

c)m (b2−a2)

2sq .units d)

m (a2−b2)2

sq .units

2. The area bounded by the line y=2x,X-axis and the ordinates x=1, x=2isa) 1 Sq.units b) 3 Sq.unitsc) 2 Sq.units d) 4 Sq.units

3. The area bounded by the line y=3x,X-axis and the ordinates x=1, x=3isa) 2 Sq.units b) 8 Sq.unitsc) 4 Sq.units d) 12 Sq.units

4. The area bounded by the line y=x,X-axis and the ordinates x=-1, x=2is [Rajastan P.E.T.2001]

a)52Sq .units b)

13Sq .units

c)32Sq .units d)

23Sq .units

5. The area bounded by the line y=mx+c,X-axis and the ordinates x=a, x=b,(a<b) isa) m (b2−a2)

2+c (b−a)sq .units

b) m (b2−a2)2

−c (b−a)sq .units

c) m (b2−a2)2

−c (a−b)sq .units

d) m (b2−a2)2

+c (a−b)sq .units

6. The area bounded by the line y=2x+3,X-axis and the ordinates x=2, x=5isa) 12 Sq.units b) 15 Sq.unitsc) 24 Sq.units d) 30 Sq.units

7. The area bounded by the line 2y=5x+7,X-axis and the ordinates x=2, x=8isa) 48 Sq.units b) 64 Sq.unitsc) 96 Sq.units d) 192 Sq.units

8. The area bounded by the line 2y=x-2,X-axis and the ordinates x=-4, x=-1isa)34Sq .units b)

94Sq .units

c)274Sq .units d)

814Sq .units

9. The area bounded by the curvey=x3,X-axis and the ordinates x=1, x=3isa) 5 Sq.units b) 10 Sq.unitsc) 20 Sq.units d) 40 Sq.unts

10. The area bounded by the curvey= 4x2

,X-axis and the ordinates x=1, x=3is

a)13Sq .units b)

23Sq .units

c)43Sq .units d)

83Sq .units

11. The area bounded by the curvey=2√1−x2,X-axis and the ordinates x=0, x=1isa)π2sq .units b)

3π2sq .units

c) π sq .units d) 3 π sq .units12. The area bounded by the curvey2(x2+6 x−55)=1X-axis and the ordinates x=7, x=14is

a) log2 sq.units b) 2log2 sq.unitsc) 3log2 sq.units d) 4log2 sq.units

13. Theareabounded by the loopof the curvey2=x2 (1−x ) is

a)25sq .unts b) 85sq.unts

c)215

sq .unts d)815

sq .unts

14. The area bounded by the curve xy=2,X-axis and the ordinates x=1, x=4isa) log 2 sq.units b) 2log 2 sq.unitsc) 3log 2 sq.units d) 4log 2 sq.units

15. The area bounded by the curve xy=4,X-axis and the ordinates x=1, x=3isa) log 3sq.units b) 2log 3sq.unitsc) 3log 3sq.units d) 4log 3sq.units

16. The area bounded by the curve xy=16,X-axis and the ordinates x=4, x=8isa) log 2 sq.units b) 2log 2 sq.unitsc) 4log 2 sq.units d) 16log 2 sq.units

17. The area bounded by the curve 2xy=1,X-axis and the ordinates x=2, x=8isa) log 2 sq.units b) 2log 2 sq.units

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c) 3log 2 sq.units d) 4log 2 sq.units18. The area bounded by the curvey=sin x,X-axis and the ordinates x=1, x=2 πis

a) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units

19. The area bounded by the curvey=sin2 x,X-axis and the ordinates x=π4 , x=3π4 is

a) 1 sq.units b) 2sq.unitsc)12sq .units d)

14sq .units

20. The area bounded by the curvey=cos x,X-axis and the ordinates x=0 tox=π2 is

a) 1 Sq.units b) 2 Sq.unitsc) 4 Sq.units d) 8 Sq.units

21. The area bounded by the curvey=cos x,X-axis and the ordinates x=0, x=πisa) 1 sq.units b) 2 sq.unitsc) 4 sq.units d) 8 sq.units

22. The area bounded by the curvey=cos x,X-axis and the ordinates x=0 tox=2 πisa) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units

23. The area bounded by the curvey=2cos x,X-axis and the ordinates x=0 to2π isa) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units

24. The area bounded by the curvey=cos3 x,0<x<π6 is

a)13sq .units b)

23sq .units

c)16sq .units d)

32sq .units

25. The area bounded by the curvey=sin2( x2 ),X-axis and the ordinates x=0 ,π6 isa) 2𝜋−2sq. units b) 4 (π−2 ) sq .units

c)12

(π−2 ) sq .units d)14

(π−2 ) sq .units

26. The area bounded by the curvey=2x+sin x,y=0 from x=0, x=π2 is

a) 1− π2

4sq .units b) 1+ π

2

4sq .units

c) 1− π2

2sq .units d) 1+ π

2

2sq .units

27. The area bounded by the curve y2=4ax and its latus rectum is [Rajastan P.E.T.1997,2000,2002]

a)4 a2

3sq .units b)

8a2

3sq .units

c)4 a√a3

sq .units d)8a√a3

sq .units

28. The area bounded by the curve y2=8 x and theline x=2 isa)43sq .units b)

83sq .units

c)163sq .units d)

323sq .units

29. The area bounded by the curve y2=16 x and theline x=4 isa)1283

sq .units b)643sq .units

c)323sq .units d)

163sq .units

30. The area bounded by the curve y2=4 x and theline x=1 ,x=4 above X-axis isa)73sq .units b)

143sq .units

c)283sq .units d)

563sq .units

31. The area bounded by the curve y2=4 x and theline x=2 ,x=4 above X-axis isa)43

(4−√2 ) sq .units b)83

(4−√2 ) sq .units

c)43

(2−√2 ) sq .units d)83


32. The area bounded by the curve y2=9 x and theline x=4,x=9 above X-axis isa) 9 sq.units b) 18 sq.unitsc) 19 sq.units d) 38 sq.units

33. The area bounded by the curve x2=ayX-axis and the lines x=a isa)a2sq . units b)

a3sq .units

c) a2

2sq .units d) a

2

3sq .units

34. The area bounded by the curve x2=ay , X-axis and the linesx=a,x=2a isP a g e |2

a) 7a2

3sq .units b) 5a

2

3sq .units

c)7a3sq .units d)

5a3sq .units

35. The area bounded by the parabola x2=4 y , the lines y=2,y=4 and Y-axis in the first quadrant isa)43

(8−2√2 ) sq .units b)83

(8−2√2 ) sq .units

c)43

(4−2√2 ) sq .units d)83

(4−2√2 ) sq .units

36. The area bounded by the parabola x2=16 y , the lines y=1,y=4 and Y-axis in the first quadrant is

a)1123


c)283sq .units d)

143sq .units

37. The area bounded by the parabola y=4 x2 , x ≥0,Y-axis & the lines the lines y=1,y=4 is

a)13sq .units b)

23sq .units

c)53sq .units d)

73sq .units

38. The area bounded by the parabola y=√6 x+4,X-axis from x=0 to x=2 in the first quadrant is

a)569sq .units b)

289sq .units

c)569sq .units d)

283sq .units

39. The area bounded by the parabola y=4√ x−1 ,1≤ x≤3and X-axis in the first quadrant is

a)2√23

sq .units b)4 √23

sq .units

c)16√23

sq .units d)32√23

sq .units

40. The area bounded by the parabola y=2x−x2,X-axis is

a)23sq .units b)

32sq .units

c)43sq .units d)

42sq .units

41. The area bounded by the parabola y=4 x−x2, X-axis is [M.P.C.E.T.1999,2003]

a)43sq .units b)

83sq .units

c)163sq .units d)

323sq .units

42. The area bounded by the parabolay=16−x2 ,0≤ x≤4 and the co-ordinate axes is

a)163sq .units b)

323sq .units

c)643sq .units d)

1283

sq .units

43. The area bounded by the parabolay=4−x2X-axis and lines x=0,x=2 is

a)163sq .units b)

83sq .units

c)43sq .units d)

23sq .units

44. The area bounded by the parabolay=x2+3X-axis and lines x=0,x=3 is


45. The area bounded by the parabolay=x2+1X-axis and lines x=0,x=3 is


46. The area bounded by the parabolay=2−x2X-axis and lines x=-1,x=1 is

a)23sq .units b)

53sq .units

c)103sq .units d)

203sq .units

47. The area bounded by the parabolay=x2−3 xthe lines y=2x and X-axis is

a)73sq .units b)

143sq .units

c)283sq .units d)

493sq .units

48. The area bounded by the parabolay=x2−3 xthe lines y=2x is

a)253sq .units b)

1253

sq .units

c)256sq .units d)

1256

sq .units

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49. The area bounded by the curve x=a t 2 , y=2atBetween the ordinates corresponding to t=1 and t=2 is

a) 7a2

3sq .units b) 14a

2

3sq .units

c) 28a2

3sq .units d) 56a

2

3sq .units

50. The area bounded by the curve y=ex,x=0,x=2 about X-axis isa) e−1 sq .units b) e−2 sq .unitsc) e2−1 sq .units d) e2−2 sq .units

51. The area bounded by the curve y=log x, X-axis and the ordinates x=1 and x=e isa) e sq .units b) 2e sq .unitsc) log e sq .units d) 2 log e sq .units

52. The area enclosed by the circle of radius r is a) πr sq .units b) 2πr sq .unitsc) π r2 sq .units d) 2π r2 sq .units

53. The area enclosed by the circle x2+ y2=9 isa) 2π sq .units b) 4 π sq . unitsc) 3 π sq .units d) 9 π sq .units

54. The area enclosed by the circle x2+ y2=16 isa) 2π sq .units b) 4 π sq . unitsc) 8 π sq.units d) 16 π sq .units

55. The area enclosed by the circle x2+ y2=2ax from x=0 to x=a above X-axis is

a) π a2

2sq .units b) π a

2

4sq .units

c) π a2

3sq .units d) π a

2

6sq .units

56. The area bounded by the ellipse b2 x2+a2 y2=a2b2 is [Karnataka C.E.T.1993]

a) πabsq .units b) 2πab sq .unitsc)π2ab sq .units d)

3π2absq .units

57. The area bounded by the ellipse 4 x2+9 y2=36 is

a) π sq .units b) 3𝜋 sq. unitsc) 6 π sq .units d) 18π sq .units

58. The area bounded by the ellipse 9x2+16 y2=144 is

a) 3 π sq .units b) 4 π sq . unitsc) 6 π sq .units d) 12π sq .units

59. The area bounded by the ellipse x2

25+ y2

16=1 is

a) 5 π sq .units b) 10π sq .unitsc) 15π sq .units d) 20 π sq .units


a2+ y

2

b2=1 and chord AB where A≡(a ,0) and B

≡(0 , b) isa)ab (π−2 )2

sq . units b)ab(π−2)

4sq .units

c)ab (π−4 )

2sq .units d)

ab(π−4)4

sq . units

61. The area bounded by the ellipse b2 x2+a2 y2=a2b2 and the ordinates x=0,x=ae,(e<1)is

a) ab (e √e2−1+sin−1e )b) ab (e√e2−1−sin−1e )c) ab (e√1−e2+sin−1 e )d) ab (e √1−e2−sin−1 e)

62. The area of region bounded by the parabola y2=4ax and the line y=x isa)8a2

3sq .units b)

4 a2

3sq .units

c)2a2

3sq .units d)

a2

3sq .units

63. The area of region bounded by the parabola y2=4 x and the line y=x isa)163sq .units b)

83sq .units

c)43sq .units d)

23sq .units

64. The area of region bounded by the parabola y2=4 x and the line y=2x isa)83sq .units b)

43sq .units

c)23sq .units d)

13sq .units

65. The area of region bounded by the parabola y=x2 and the line y=x is [U.P.S.E.A.T. 2004]

a)12sq .units b)

13sq .units

c)14sq .units d)

16sq .units

66. The area of region bounded by the parabola y2=4 x and the line x+ y=2 isP a g e |4

a)12sq .units b)

32sq .units

c)72sq .units d)

92sq .units

67. The area of region bounded by the parabola y2=2 x and the line y=4 x−1isa)316


c)916

sq .units d)932sq .units

68. The area of region bounded by the parabola y2=4 x and the line 3 y=2 x+4isa)12sq .units b)

13sq .units

c)23sq .units d)

32sq .units

69. The area of region bounded by the parabola y2=x,X-axis and the line x+ y=2 in the first quadrant isa)72sq .units b)

73sq .units

c)75sq .units d)

76sq .units

70. The area of region bounded by the parabola y2=16 x,and the chord AB,where A≡(1,4) and B≡(9,12)

a)23sq .units b)

43sq .units

c)83sq .units d)

163sq .units

71. The area of region bounded by the parabola 4 y=3 x2 and the line 2y=3 x+12isa) 27 sq.units b) 18 sq.unitsc) 9 sq.units d) 3 sq.units

72. The area of region bounded by the parabola x2= y and the line y=x+2 and X-axisisa)32sq .units b)

23sq .units

c)92sq .units d)

29sq .units

73. The area of region bounded by the parabola y=x2+1 and the line y=2¿) and X-axisisa)23sq .units b)

43sq .units

c)83sq .units d)

163sq .units

74. The area of region bounded by the parabola y=x2+1 and the line y=x , x=0 , x=2 isa)83sq .units b)

43sq .units

c)23sq .units d)

13sq .units

75. The area of region bounded by the parabola y=x2+2 and the line y=x , x=0 , x=3 isa)72sq .units b)

172sq .units

c)212sq .units d)

152sq .units

76. The area of region bounded by the parabola y=x2+1 and the line y=x+1 isa)12sq .units b)

13sq .units

c)16sq . units d)

112sq .units

77. The area of region bounded by the parabola y=x2−6x+11 and the line y=5−x isa)12sq .units b)

13sq .units

c)16sq . units d)

112sq .units

78. The area of region bounded by the parabola y2=4ax and x2=4 ay isa) 2a

2

3sq .units b) 4 a

2

3sq .units

c) 8a2

3sq .units d) 16a

2

3sq .units

79. The area of region bounded by the parabola y2=x and x2= y is [M.P.C.E.T.1997]

a)163sq .units b)

83sq .units

c)13sq .units d)

43sq .units

80. The area of region bounded by the parabola y2=4 x and x2=4 y is [Karnataka C.E.T 1999,2003]

a)23sq .units b)

43sq .units

c)83sq .units d)

163sq .units

81. The area of region bounded by the parabola y2=12 x and x2=12 y isa) 3 sq.units b) 4 sq.unitsc) 36 sq.units d) 48 sq.units

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82. The area of region bounded by the parabola 4 y2=9 x and 3 x2=16 y isa) 4 sq.units b) 2 sq.unitsc) 6 sq.units d) 3 sq.units

83. The area of region bounded by the parabola y2=x+1 and y2=−x+1 isa)23sq .units b)

43sq .units

c)83sq .units d)

163sq .units

84. The area of the region in the first quadrant enclosed by the circle x2+ y2=32 and the line y=x isa) π sq .units b) 2π sq .unitsc) 3 π sq .units d) 4 π sq . units

85. The area of the region enclosed by the circle x2+ y2=36 and the line x+ y=6 isa) 9 π+18 sq .units b) 9 π−18 sq .unitsc) 3 π+6 sq .units d) 3 π−6 sq .units

86. The area of the smaller region bounded by the ellipse 4 x2+9 y2=36 and the line 2 x+3 y=6 is

a)32

(π−1 ) sq .units b)32

(π−2 ) sq .units

c)34

(π−1 ) sq .units d)34

(π−2 ) sq .units

87. The area bounded by the curves y=x2+5 and y=x3∧thelines x=1 , x=2 isa)3112sq .units b)

4312

sq .units

c)316sq .units d)

436sq .units

88. The area of the triangular region bounded by the line2 y=x , y=2x and x=4 isa) 3 sq.units b) 6 sq.unitsc) 12 sq.units d) 24 sq.units

89. The area of the triangular region bounded by the liney=2x+1 , y=3x+1 and x=4 isa) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units

90. The area of ∆ ABC with vertices A(2,1),B(3,1) and C(5,7) isa) 5 sq.units b) 10 sq.unitsc) 15 sq.units d) 20 sq.units

91. The area of ∆ ABC with vertices A(2,1),B(3,4) and C(5,2) isa) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units

92. The area of ∆ ABC with vertices A(2,0),B(4,5) and C(6,3) isa) 3 sq.units b) 4 sq.unitsc) 7sq.units d) 8 sq.units

93. The area of the region enclosed by the curves y=sin x andy=cos x from x=0¿ x= π2 and X-axis is

a) 2−√2 sq .units b) 1−√2 sq .unitsc) 2−2√2 sq .units d) 1−2√2 sq .units

94. The area of the region enclosed by the curves y=sin x andy=cos x and Y-axis isa) 1−√2 sq .units b) 2−√2 sq .unitsc) −1+√2 sq .units d) −2+√2 sq .units

95. The volume of solid generated by revolving the area bounded by the line y=x , y=0 , y=2 and Y-axis about Y-axis is

a)2π3cu .units b)

4 π3cu .units

c)8π3cu .units d)

16π3

cu .units

96. The volume of solid generated by revolving the area bounded by the line y=x , y=0 , x=4 about X-axis is

a)64 π3

cu .units b)32π3

cu .units

c)16π3

cu .units d)8π3cu .units

97. The volume of solid generated by revolving the area bounded by the line y=2x , x=0 , x=3 and X-axis about X-axis is

a) 9 π cu .units b) 18π cu .unitsc) 27 π cu .units d) 36 π cu .units

98. The volume of solid generated by revolving the area bounded by the line y=x+1 , X−axis∧the lines x=0 , x=2 about x-axis is

a)13π3

cu .units b)26π3

cu .units

c)10π3

cu .units d)20π3

cu .units

99. The volume of solid generated by revolving the area bounded by the curves xy=2 , Y−axis∧lines y=1, y=4 about Y-axis is

a) π cu .units b) 2π cu .unitsc) 3 π cu .units d) 4 π cu .units

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100. The volume of solid generated by revolving the area bounded by the curves xy=1 ,∧lines x=1 , x=2 about X-axis is

a)π2cu .units b)

3π2cu .units

c) π cu .units d) 3 π cu .units101. The volume of solid generated by revolving the area bounded by the curves y=simx ,¿x=0 , x= π

2 about X-axis isa)π2cu .units b)

π4cu .units

c) π2

2cu .units d) π

2

4cu .units

102. The volume of solid generated by revolving the area bounded by the parabola y2=4ax,X-axis and its latus rectum about X-axis is

a) π a3cu .units b) 2π a3cu .unitsc) 3 π a3 cu .units d) 4 π a3cu .units

103. The volume of solid generated by revolving the area bounded by the parabola y2=16 x,X-axis and its latus rectum about X-axis is

a) 128π cu .units b) 64 π cu .unitsc) 32π cu .units d) 16 π cu .units

104. The volume of solid generated by revolving the area bounded by the parabola y2=16 x,and the lines x=0 , x=2 about X-axis is

a) 8 π cu .units b) 16 π cu .unitsc) 24 π cu .units d) 32π cu . units

105. The volume of solid generated by revolving the area bounded by the parabola y2=4 x,and the lines x= y about X-axis is

a)32π3

cu .units b)16π3

cu .units

c)8π3cu .units d)

4 π3cu .units

106. The volume of solid generated by revolving the area bounded by the parabola y2=4 x,and the lines y¿2 x about X-axis is

a)π2cu .units b)

π3cu .units

c)2π3cu .units d)

3π2cu .units

107. The volume of solid generated by revolving the area bounded by the parabola y2=3 x,and the lines y¿ x about X-axis is

a)3π2cu .units b)

9π2cu .units

c)27π2

cu .units d)81π2

cu .units

108. The volume of solid generated by revolving the area bounded by the parabola x2= y,and the lines y¿ x about X-axis is

a)π5cu .units b)

2π5cu .units

c)π15cu . units d)

2π15

cu .units

109. The volume of solid generated by revolving the area bounded by the parabola x2= y,and the lines y¿2 x about X-axis is

a)4 π15

cu .units b)16π15

cu .units

c)64 π15

cu .units d)256π15

cu .units

110. The volume of solid generated by revolving the area bounded by the parabola x2=4 y,X-axis and the lines x¿−3and x=4 about X-axis is

a)1267π80

cu .units b)781π80

cu .units

c)1267π20

cu .units d)781π20

cu .units

111. The volume of solid generated by revolving the area bounded by the parabola y2=16 x,and chord AB,A≡ (1,4 ) ,B≡(9,12) about X-axis is

a)4 π3cu .units b)

16π3

cu .units

c)64 π3

cu .units d)256π3

cu .units

112. The volume of solid generated by revolving the area bounded by the parabola y2=4 x,and x2=4 y,about X-axis is

a)96π5

cu .units b)48 π5

cu .units

c)24 π5

cu .units d)12π5

cu .units

113. The volume of solid generated by revolving the area bounded by the parabola y2=x,and x2= y,about X-axis is

a)π5cu .units b)

3π5cu .units

c)π15cu . units d)

3π15

cu .units

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114. The volume of solid generated by revolving the area bounded by the circle x2+ y2=25,and the line 3 x=4 y and X-axis in first quadrant is

a)5π3cu .units b)

10π3

cu .units

c)25π3

cu .units d)50π3

cu .units

115. The volume of solid generated by revolving the area bounded by the circle x2+ y2=36,and the line x+ y=6 and X-axis in first quadrant is

a) 18π cu .units b) 36 π cu .unitsc) 54 π cu .units d) 72π cu .units

116. The volume of solid generated by revolving the area bounded by the circle x2+ y2=25,and the line x=0 , x=3 and X-axis is

a) 26 π cu .units b) 38 π cu .unitsc) 52π cu .units d) 66 π cu .units

117. The volume of solid generated by revolving the area bounded by the circle x2+ y2=4,and the parabolay2=3 x and X-axis in first quadrant about X-axis is

a)19π2

cu .units b)19π3

cu .units

c)19π6

cu .units d)19π12

cu .units

118. The volume of solid generated by revolving the area bounded by the ellipse x2

a2+ y

2

b2=1 about major axis is

a) 2π a2b3

cu .units b) 4 π a2b3

cu .units

c) 2π b2a3

cu .units d) 4 π b2a3

cu .units

119. The volume of solid generated by revolving the area bounded by the ellipse 9 x2+16 y2=144 about major axis is

a) 12π cu .units b) 24 π cu .unitsc) 36 π cu .units d) 48 π cu .units

120. The volume of solid generated by revolving the area bounded by the ellipse x2

16+ y2

9=1 about major axis is

a) 32π cu .units b) 64 π cu .units

c)72π3

cu .units d)144π3

cu .units

121. The volume of solid generated by revolving the area bounded by the ellipse 9 x2+4 y2=36 about Y-axis is

a) 2π cu .units b) 4 π cu .unitsc) 8 π cu .units d) 16 π cu .units

122. The volume of solid generated by revolving the area bounded by the arc AB and chord AB of ellipse x2

a2+ y

2

b2=1 with AA’ as

major axis and BB’ as minor axis isa) πab

2

3cu .units b) π a

2b3

cu . units

c) πab2

6cu .units d) π a

2b6

cu . units

123. The volume of solid generated by revolving the area bounded by the one branch of rectangular hyperbola x2− y2=a2 about X-axis isa)π a3

3cu .units b)

π a3

2cu .units

c)4 π a3

3cu .units d)

2π a3

3cu .units

124. The volume of sphere of radius r isa)π r3

3cu .units b)

π r3

2cu .units

c)4 π r3

3cu .units d)

2π r3

3cu .units

125. The volume of sphere of radius 4 isa)256π3

cu .units b)128π3

cu .units

c)64 π3

cu .units d)32π3

cu .units

126. The volume of right circular cylinder of base radius r and height h isa) πrhcu .units b) 2πrhcu .unitsc) π r2hcu .units d) 2π r2hcu .units

127. The volume of right circular one of base radius r and height h isa) π h

2 r2

cu .units b) π h2 r3

cu .units

c) π r2h2

cu .units d)

128. The area bounde by the X-axis and The curve y=sin x and x=0 , x=π is [Kerala P.E.T. 2002]


P a g e |8

129. The area bounded by the curve y2=x,line y=4,Y-axis is [Roorkee 1995,Rajastan P.E.T.2003]

a)163sq .units b)

643sq .units

c) 7√2 sq .units d) 8√2 sq .units130. Area bounded by the curve y=x2,X-axis and linex=1,is

a)13sq .units b)

12sq .units

c) 1 sq .units d) None of these131. Area bounded by the curve y=|x|, and y¿4−|x|,is


132. Area bounded by x=1 , x=2 , xy=1 and X-axis isa) ( log 2 ) sq .units b) 2 sq .unitsc) 1 sq.units d) None of these

133. Area bounded by the curve y=2x−x2, and X-axis isa)23sq .units b) 1 sq.units

c) 2 sq.units d)43sq .units

134. Area bounded by the curve y2=16 x, and line y=mx is 23 ,then m is equal to

a) 3 b) 4c) 1 d) 2

135. Area bounde by the curves y2=x∧x2= y isa)23sq .units b) 1 sq.units

c)12sq .units d) None of these

136. Area of the smaller portion bounded by the ellipse 16 x2+25 y2=400 and line 4 x+5 y=20 isa) 20 π sq .units b) 5 (π−2 ) sq .unitsc) 10 (π−2 ) sq .units d) None of these

137. Area enclosed within the curve |x|+|y|=1 isa) 1 sq.units b) 2 sq.unitsc) 4 sq.units d) None of these

138. The area bounded by Y-axis,y=cos x∧ y=sin x , x ≥0 is

a) 2 (√2−1 ) sq .units b) (√2−1 ) sq .unitsc) (√2+1 ) sq .units d) √2 sq .units

139. Area enclosed by y=1∧±2 x+ y=2 (in sq.unts) isa)12 b)

14

c) 1 d) None of these140. Area common to curve y=x3∧ y=√ x is

a)512 b)

53

c)54 d) None of these

141. The area of the region bounded by y=|x−1|∧¿y=1 is

a) 1 b) 2c)12 d) None of these

142. The area of the region bounded by the curve y2=4 x Y-axis and the line y=3 is

a) 2 sq.units b)94sq .units

c) 6√3 sq .units d) None of these143. The area bounded by the curve y ,=|x|,y=|x−1|∧¿ X-axis is



144. Area of rhombus enclosed by the lines ax ±by±c=0 is

a)12sq .units b)

12c2

absq .units

c) 2c2

absq .units d) None of these

145. The area of the quadrilateral formed by the tangents as the end points of the latus rectum to the ellipse x2

9+ y

2

5=1 is

a)274sq .units b) 9 sq.units


146. The area bounded by y=log x,X-axis and ordinates x=1 , x=2 isa)12

(log 2 )2 sq . units b) log2esq .units

P a g e |9

c) log 4esq .units d) None of these

147. The area bounded by y=x,X-axis and ordinates x=−1 , x=2 isa) 0 sq.units b)

12sq .units

c)32sq .unit d)

52sq .units

148. The area between the parabola y=x2 and the line y=x isa)16sq .units b)

13sq .units


149. The area of region bounded by the curvesy=|x−2|, x=1, x=3 and X-axis isa) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units

150. The area bounded by the curves y=√x ,2 y+3=x and X-axis in Ist Quadrant is


c) 36 sq.units d) 18 sq.units151. Area bounded by the curve y=sin x and the linesx=0 , x=π

2∧x=−π

2 isa) 2 b) 4c) 8 d) 16

152. Volume of solid obtained by the complete revolution of the ellipse x2

a2+ y

2

b2=1 , a>b ,

About major axis isa)43π a3 b)

43π b3

c)43π ab2 d)

43π a2b

153. Volume of solid surface generated by rotating the curve y=4 x2 from(0,0 )¿(1,4 ) about Y-axis isa) π cu .units b) 2π cu .unitsc) 4 π cu . units d) 8 π cu .units

154. The volume of the solid generated by revolving the area enclosed by y=4 x2 , x=0∧ y=16About Y-axis is

a) 16 π cu .units b) 32π cu . unitsc) 64 π cu .units d) 48 π cu .units

155. The volume of the solid generated by revolving the area boundedby the parabola x2=4 y,X-axisThe line x=−3∧x=4 about X-axis is

a) 15.783π c .u . b) 15.873π c .u .c) 15.837 π c .u . d) 15.738π c .u .

156. The volume of the solid generated by revolving the region bounded by the curve x= 2

y ,Y-axis x=0The line y=1∧ y=4 about Y-axis isa) 2π cu .units b) 3 π cu .unitsc) 4 π cu .units d) 5 π cu .units

157. The triangle bounded by the lines y=0 , y=x∧¿x=0 is revolved X-axis .Then the volume of the solid thus generated is

a)64 π3 b)

16π3

c)12π3 d)

4 π3

158. Area enclosed between the curves y=x1 /3 ,the Y-axis and the lines y=−1, y=1 isa) 0 b)

12

c)32 d)

33

159. The area of the triangle formed by the lines y=2x , x=0∧ y=2 isa) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units

160. The area of the region bounded by the curve y2=16 x and lines x=4 , x=1 above the X-axis isa)365sq .units b)

473sq .units

c)563sq .units d)

597sq .units

161. The area between parabola y2=4 x and its latus rectum isa)23sq .units b)

83sq .units

c)163sq .units d)

323sq .units

162. The area bounded by y=sin x,the X-axis and the ordinates x=0 , x=2π ,isa) 4 sq.units b) 6 sq.unitsc) 8 sq.units d) 10 sq.units

P a g e |10

163. The area of region bounded by the curves y=x2+2 , y=x , x=0∧x=3is

a) 8.5 sq.units b) 9.5 sq.unitsc) 10.5 sq.units d) 11.5 sq.units

164. The area enclosed by two curves y2=x+1∧¿y2=−x+1,is

a)23sq .units b)

35sq .units

c)43sq .units d)

83sq .units

165. The area of the region bounded by the parabola y=4 x2 the Y-axis ,y=1∧ y=4,lying in the first quadrant isa)83sq .units b)

73sq .units

c)53sq .units d)

13sq .units

166. The area bounded by the curve y=log x,the axis of X and the line x=a , x=b isa) a log( be )+b log( ae ) b) a log( be )−b log( ae )c) −a log( be )+b log( ae ) d) None of theses

167. The area bounded by the curve y=cos x¿0¿ π,is


168. The area of region in the first quadrant bounded by the circle x2+ y2=32,the line y=x and X-axis isa) 4 π sq .units b) 3 π sq .unitsc) 2π sq .units d) 1π sq .units

169. The volume of the solid generated by the revolving the region bounded by the parabola y2=16 x and its latus rectum is about X-axis is


170. The volume obtained by the revolving the region bounded by x2− y2=9 , y=0 and its latus rectum is about X-axis isa) 18π (2−√2 )c .u b) 20 π (2−√2 )c .uc) 22π (2−√2 )c .u d) 32π (2−√2 ) c .u

171. Find the volume of solid generated by the revolving the region bounded by the line y=2x,X-axis and the lines x=0 , x=3


172. Volume of solid generated by the revolving the curve about X-axis ,if the area enclosed by the parabola y2=4ax and its latus rectum isa) 4 πacub .units b) 4 π a2cub .unitsc) 4 π a3cub .units d) None of these

173. Find the area of the triangular region whose side have the equationy=2x+1 , y=3x+1∧¿x=4


174. The circle x2+ y2=a is revolved about X-axis.Find the volume of sphere so formeda)43π a2 cu .units b)

43πacu .units

c)43π a3 cu .units d) None of these

175. The area bounded by the curve y=x,X-axis and lines x=−1 , x=2,is given bya) 2/3 b) 5/2c) 7/4 d) 9/11

176. The area of ellipse x2a2

+ y2

b2=1,is given by

a) πab b) 2πab

c)π2ab d)

23πab

177. The line y=x+1 is revolved about X-axis.The volume of solid of revolution formed by revolving the area of covered by the given curve,x-axis and the lines x=0 , x=2,isa)13π3 b)

17π3

c)19π3 d)

26π3

178. The surface of the sphere obtained by revolving the circle x=r . cosθabout X-axis is equal yoa) π r2 b) 2π r2

c) 4 π r2 d)43πr 2

P a g e |11

179. Find the volume of the solid formed when the area between the X-axis ,the line x=2 , x=4 andThe curve y=x2 is rotated once about the X-axis.Leaves your answer as multiple of π.

a)7723

π cu .units b)8825

π cu .units

c)9923

π cu .units d) None of these180. If the portion of the parabola y2=4 x lying between the vertex and the latus rectum is revolved about X-axis, then the volume of the solid thus generated is

a)π2 b) π

c) 2π d) π2

181. The area bounded by the curve y=log x,X-axis and the ordinates x=1∧x=e is a) e sq .units b) 2e sq .unitsc) log e sq .units d) 2 log e sq .units

182. The area bounded by the curve y2=4 x and the line x=1 , x=4 above X-axis isa)73sq .units b)

143sq .units

c)283sq .units d)

563sq .units

183. The area of the region bounded by the parabolay2=x2+1 and the line y=x+1 is

a)12sq .units b)

13sq .units

c)16sq .units d)

112sq .units

184. The area bounded by the parabolay2=4ax andIts latus rectum is

a)4 a2

3sq .units b)

8a2

3sq .units

c)4 a√a3

sq .units d)8a√a3

sq .units

185. The area of triangular region bounded by the linesy=2x+1 , y=3x+1∧x=4is


186. The area bounded by the parabola y=x2+3,

X-axis and the lines x=0 , x=3 isa) 3 sq.units b) 6 sq.unitsc) 12 sq.units d) 18 sq.units

187. The area bounded by the line y=x,X-axis and theOrdinates x=−1 , x=2 isa)52sq .units b)

32sq .units

c)13sq .units d)

23sq .units


a2+ y

2

b2=1 and

Chord AB where A≡(a ,o) and B≡(0 , b) isa)ab (π−2 )2

sq . units b)ab(π−2)

4sq .units

c)ab (π−4 )

2sq .units d)

ab(π−4)4

sq . units

189. The area of the region bounded by the parabolay2=4ax and the line y=mx is

a)8 a2

3m3sq .units b)

4 a2

3m3sq .unitsl

c)2a2

3m3sq .units d)

a2

3m3sq .units

190. The area bounded by the curve x=at 2 , y=2atBetween the ordinates corresponding to t=1∧t=2 is

a) 7a2

3sq .units b) 14a

2

3sq .units

c)28a2

3sq .units d)

56a2

3sq .units

191. The area bounded by the curve y=sin2 x ,X-axisAnd the lines x=π

4∧x=3 π

4 isa) 1 sq.units b) 2 sq.unitsc)12sq .units d)

14sq .units

192. The area bounded by the curve y2=kx,X-axis and the ordinates x=c isa)c3 √kc sq .units b)

2c3 √kc sq .units

c)4 c3 √kc sq .units d)

8c3 √kc sq .units

193. The area bounded by the line y=x3,X-axis and the ordinates x=1 , x=3 isP a g e |12


194. The area bounded by the curve y=cos3 x ,

0≤ x≤ π6 is

a)13sq .units b)

23sq .units

c)16sq .units d)

32sq .units

195. The area enclosed by the circle of radius r is a) πr sq .units b) 2πr sq .unitslc) π r2 sq .unitsl d) 2π r2 sq .units

196. The volume of solid generated by revolving the area bounded by the lines y=x , y=0 , y=2 and Y-axis is about Y-axis is

a)2π3cu .units b)

4 π3cu .units

c)8π3cu .units d)

16π3

cu .units

197. The volume of right circular cylinder of base radius r and height h isa) πrhcu .units b) 2πrhcu .unitsc) π r2hcu .units d) 2π r2hcu .units

198. The volume of right circular one of base radius r and height h isa)πr h2

2cu .units b)

πr h2

3cu .units

c)π r2h2

cu .units d)π r2h3

cu .unitsl

199. The volume of solid generated by revolving the area bounded by the lines y=2 x , x=0 , x=3 and X-axis about X-axis is

a) 9 π cu .units b) 18π cu .unitsc) 27 π cu .units d) 36 π cu .units

200. The volume of solid generated by revolving the area bounded by the parabola x2=4 y,X-axis and lines x=−3∧x=4 about X-axis is

a)1267π80

cu .units b)781π80

cu .units

c)1267π20

cu .units d)781π20

cu .units

201. The volume of solid generated by revolving the area bounded by the circle x2+ y2=25 ,th lines x=0 , x=3 about X-axis is


202. The volume of solid generated by revolving the area bounded by the one branch of rectangular hyperbola x2− y2=a2 line x=2a about X-axis isa)π a3

3cu .units b)

π a3

2cu .units

c)4 π a3

3cu .units d)

2π a3

3cu .units

203. The volume of solid generated by revolving the area bounded by the curve y=sin x ¿ x=0¿ x=π

2 cu.units about X-axis isa)π2sq .units b)

π4sq .units

c) π2

2sq .units d) π

2

4sq .units

204. The volume of solid generated by revolving the area bounded by the circle x2+ y2=4 and parabola 2 y=3 x and X-axis in first quadrant about X-axis is

a)19π2

cu .units b)19π3

cu .units

c)19π6

cu .units d)19π12

cu .units

Application of Definite Integrals205. The volume of solid generated by revolving the area bounded by the curve xy=1 ,and the linesx=1∧x=4 about X-axi is

a)π2cu .units b)

3π2cu .units

c) π cu .units d)3π4cu .units

P a g e |13

SAMEJ TUTORIALSMT-CET 2013

Date : TEST ID: 33Time : 06:09:00 Hrs. MATHEMATICS-IIMarks : 410


: ANSWER KEY :

1) c 2) c 3) d 4) a5) a 6) d 7) c 8) c9) c 10) d 11) a 12) a13) d 14) d 15) d 16) d17) a 18) b 19) a 20) a21) b 22) b 23) c 24) a25) d 26) b 27) b 28) d29) a 30) c 31) b 32) d33) d 34) a 35) a 36) b37) d 38) a 39) c 40) c41) d 42) d 43) a 44) d45) d 46) c 47) d 48) d49) d 50) c 51) c 52) c53) d 54) d 55) b 56) a57) c 58) d 59) d 60) b61) c 62) a 63) b 64) d65) d 66) d 67) d 68) b69) d 70) d 71) a 72) c73) b 74) a 75) c 76) c77) c 78) d 79) c 80) d81) d 82) a 83) c 84) d85) b 86) a 87) b 88) c89) c 90) b 91) d 92) c93) a 94) c 95) c 96) a97) d 98) b 99) c 100) a101) d 102) b 103) a 104) d

105) a 106) c 107) b 108) d109) c 110) a 111) d 112) a113) d 114) d 115) d 116) d117) c 118) d 119) d 120) d121) d 122) a 123) c 124) c125) a 126) c 127) d 128) d129) b 130) a 131) d 132) a133) d 134) b 135) d 136) b137) b 138) b 139) a 140) a141) a 142) b 143) c 144) c145) d 146) c 147) d 148) a149) a 150) a 151) a 152) c153) b 154) b 155) c 156) b157) a 158) b 159) a 160) c161) b 162) a 163) c 164) d165) b 166) c 167) a 168) a169) c 170) a 171) b 172) c173) b 174) c 175) b 176) a177) d 178) c 179) c 180) c181) c 182) c 183) c 184) b185) c 186) d 187) a 188) b189) a 190) d 191) a 192) b193) c 194) a 195) c 196) c197) c 198) d 199) d 200) a201) d 202) c 203) d 204) c205) d

P a g e |14

SAMEJ TUTORIALSMT-CET 2013

Date : TEST ID: 33Time : 06:09:00 Hrs. MATHEMATICS-IIMarks : 410


: HINTS AND SOLUTIONS :

1 (c)The bounded area is shown in figureRequired area isA=∫

a

b

ydx

A=∫a

b

mxdx

A=[mx22 ]baA=

m (b2−a2 )2

sq .units

2 (c)

Area=∫1

2

2xdx

Area=[ x2 ]21

Area=3Sq .units

3 (d)

Area=∫1

3

3xdx

Area=32

[ x2 ]31

Area=12Sq .Units

4 (a)The bounded area is shown in figureRequired area isA=|∫

−1

0

ydx|+∫02

ydx

A=|∫−1

0

xdx|+∫02

xdx

A=|[ x22 ] 0−1|+[ x22 ]20A=1

2+2

A=52Sq .units

5 (a)The bounded area is shown in figureRequired area isA=∫

a

b

ydx

P a g e |15

A=∫a

b

(mx+c)dx

A=[mx22 +cx ]baA=

m (b2−a2 )2

+c (b−a)sq .units

6 (d)

Area=∫2

5

(2 x+3)dx

Area=[ x2+3 x ]52

Area=30Sq .units

7 (c)

Area=12∫28

(5 x+7)dx

Area=12 {52 x2+7 x}82

Area=12 {(160+56−24 ) }

Area=96Sq .units

8 (c)The bounded area is shown in figure

Required area isA=|∫

−4

1

( x2−1)dx|A=|[ x24 −x ]−1−4|A=|14 +1−4−4|A=27

4Squnits

9 (c)The bounded area is shown in figureRequired area isA=∫

1

3

x3dx

A=[ x 44 ]31A=1

4[81−1]

A=20Sq .units

10 (d)Required area isA=∫

a

b

ydx

A=∫1

3

( 4x2 )dxA=[−4x ]31A=−4

3+4

P a g e |16

A=83Sq .unts

11 (a)Required area isA=∫

0

1

2√1−x2dx

A=2[ x2 √1−x2+12sin−1 x]10

A=0+sin−1 (1 )−0−sin−10

A=π2sq .units


7

14 dx√ x2+6 x−55

A=∫7

14 dx

√ ( x+3 )2−82

A=[ log|( x+3 )+√( x+3 )2−82|]147

A=log|17+√196+84−55|−log|10+√49+42−55|A=log 32−log16A=log 2 sq. units

13 (d)The bounded area is shown in fig.The curve isy2=x2(1−x )Put y=0x2 (1−x )=0x=0 , x=1A=(1,0)Required Area isA=2∫

0

1

x √1−xdx

By∫0

a

f ( x )dx=¿∫0

a

f (a−x)dx¿

A=2∫0

1

(1−x )√ xdx

A=2[2 x√ x3 −2 xx √x5 ]10

A=43−45

A= 815

sq .units

P a g e |17

14 (d)

Area=∫1

4 2xdx

Area=2 [ log x ] 41

Area=4 log2 sq .units

15 (d)

Area=∫a

b

ydx

Area=∫1

3

( 4x )dxArea=4 [ log x ]3

1Area=4 log3 sq .units

16 (d)

Area=∫4

8

( 16x )dxArea=16 [ log x ] 8

4Area=16 log 2 sq .units

17 (a)

Area=∫2

8

( 12x )dxArea=1

2[ log x ] 8

2Area=log 2 sq .units

18 (b)The bounded area is shown in fig.The curve isA=2 AreaOAB

A=2∫0

π

sin x dx

A=2 [−cos x ]π0

A=2(−cos π+cos 0)A=4 sq . units

19 (a)The bounded area is shown in fig.The curve is

P a g e |18

A=∫π4

π2

ydx+∫π2

3 π4

− ydx

A=∫π4

π2

sin 2xdx−∫π2

3 π4

sin 2 xdx

A=[−cos2 x2 ]π2

π4−[−cos 2x2 ]

3π4π2

A=−12

[cos π−cos π2−cos 3π

2+cosπ ]

A=−12

[−1−0−0−1]

A=1Sq .units

20 (a)

Area=∫0

π2

cos xdx

Area=[sin x ]π20

Area=1Sq .unts

21 (b)

Area=∫0

π2

(cos x )dx+∫π2

π

|cos x|dx

Area=[sin x ]

π2

0+|[sin x ] ππ2 |Area=1+|1|Area=2 sq .units

22 (b)

Area=4∫0

π2

(cos x )dx

Area=4 [sin x ]π20

Area=4 sq .units

23 (c)The bounded area is shown in fig.The curve isA=4 AreaOAB

A=4∫0

π2

ydx

A=4∫0

π2

2cos xdx

A=8[sin x ]π20

A=8[sin( π2 )−sin 0]P a g e |19

A=8 sq .units 24 (a)The bounded area is shown in fig.The curve isA=∫

0

π6

cos3 xdx

A=[ sin3 x3 ] π60

A=13 [sin( π2 )−sin 0]

A=13sq .units

P a g e |20

25 (d)Required Area isA=∫

0

π2

sin2( x2 )dx

A=∫0

π2

( 1−cos x2 )dxA=1

2[x−sin x ]

π20

A=12 [ π2−sin (π2 )−0]

A=14

(π−2 ) sq .units

26 (b)

Area=∫0

π2

(2 x+sin x )dx

Area=[ x2−cos x ]π20

Area=( π24 −0)−(0−1)

Area=1+ π2

4sq .units

27 (b)The bounded area is shown in fig.The curve isA=2 AreaOSL

A=2∫0

a

2√ax dx

A=4 √a [ 2 x√ x3 ]a0A=8

3 √a[a√a−0]

A=8 a2

3sq .units

28 (d)The bounded area is shown in fig.The curve isA=2 AreaOSL

A=2∫0

2

2√2x dx

A=4 √2[ 4√23 ]

A=323sq .units

29 (a)

Area=2∫0

4

(4 √x )dx

Area=8[ 23 x32 ]40

Area=1283

sq .units

30 (c)The bounded area is shown in fig.The curve isA=∫

1

4

2√x dx

A=[ 4 x √x3 ]41

A= 43[8−1]

P a g e |21

A=283sq .units

31 (b)

Area=∫2

4

2√x dx

Area=2[ x32

32 ] 42

Area=43(8−2√2)

Area=83


32 (d)

Area=∫4

9

3√x dx

Area=3( 23 ) [ x32 ]94

Area=2(27−8)Area=38 sq .units

33 (d)The bounded area is shown in fig.The curve isA=∫

0

a

ydx

A=∫0

a x2

adx

A=[ x33a ] a0A=a2

3sq .units

34 (a)

Area=∫a

2a x2

adx

Area=[ x33a ]2aaArea=7 a

2

3sq .units

P a g e |22

35 (a)The bounded area is shown in fig.The curve isA=∫

2

4

2√ y dy

A=2[2 y √ y3 ]42

A=43

(4 √4−2√2 )

A=43

(8−2√2 ) sq .units

36 (b)

Area=∫1

4

4 √ y dy

Area=83

[ y 32 ]41

Area=563sq .units

37 (d)

Area=∫1

4 √ y2dy

Area=13

[ y 32 ]41

Area=73sq .units

38 (a)The curve is y=√6 x+4y2=6 x+4

y2=6 (x+ 23 ) represented a parabola with vertex at (−23 ,0),The bounded area is shown in fig.The curve isA=∫

0

2

√6 x+4dx

A=[ 2 (6 x+4 )32

3 (6 ) ]20A=1

9[ (16 )

32−(4 )

32 ]

A=569sq .units

39 (c)The curve is y=4√ x−1y2=16 ( x−1 ) represented a parabola with vertex at (1,0 ),

P a g e |23

The bounded area is shown in fig.The curve isA=∫

1

3

4 √x−1dx

A=4 [ 2 ( x−1 )32

3 ]31A=8

3(2√2−0)

A=16√23

sq .units

40 (c)The curve is y=2x−x2

y=−(x2−2x+1 )+1y−1=−( x−1 )2 represented a parabola with vertex at (1,1 ),The bounded area is shown in fig.The curve isA=∫

0

2

(2 x−x2)dx

A=[ x2− x3

3 ]2oA=4−8

3−0

A=43sq .units

41 (d)The curve is y=4 x−x2

y=−(x2−4 x+4 )+4y−4=−( x−2 )2 represented a parabola with vertex at (2,4 ),The bounded area is shown in fig.The curve isA=∫

0

4

(4 x−x2)dx

A=[2 x2− x3

3 ]4oA=32− 64

3

A=323sq .units

42 (d)The curve isy=16−x2

y−16=−x2 represented a parabola with vertex at (0,16 ),The required area isA=∫

0

4

(16−x2)dx

A=[16 x− x3

3 ]4oA=64−64

3

A=1283

sq .units

43 (a)

P a g e |24

The bounded Area is as shown in figureThe Required area isA=∫

0

2

(4−x2)dx

A=[4 x− x3

3 ]20A=8−8

3−0

A=163sq .units

44 (d)

Area=∫0

3

(x2+3 )dx

Area=[ x33 +3 x]30Area=18 sq .units

45 (d)

Area=∫0

3

(x2+1 )dx

Area=[ x33 +x ]30Area=12 sq .units

46 (c)

Area=2∫0

1

(2−x2 )dx

Area=2[2x− x3

3 ]10Area=2( 53 )Area=10

3sq .units

47 (d)The bounded area is shown as followCurves arey=x2−3 xand y=2xSolving themx=0 , x=5y=0 , y=10A≡ (5,10 ) ,C≡(5,0)Put y=0 inparabola equtionx (x−3 )=0x=0,3B≡(3,0)Required area is A=Area OABA=Area OAC – Area ABCA=∫

0

5

y linedx−∫3

5

y paraboladx

A=∫0

5

2xdx−∫3

5

(x2−3 x)dx

P a g e |25

A=[ x2 ]5

0−[ x33 −3x2

2 ]53

A=25−1253

+ 752

+9−272

A=493sq . units

48 (d)The bounded area is shown in figure of question 47Required area isA=Area OAB +Area ODBArea ODB=493 sq.unitsAreaODB=|∫0

3

(x2−3 x)dx|=|[ x33 −3 x2

2 ]30|AreaODB=|9−272 |=92Area=49

3+ 92

Area=1256

sq .units

49 (d)The curves isx=a t 2 and y=2aty2=4a2t 2

y2=4a (a t2)y2=4ax represent a standard parabolaAt t=1 , x=aAt t=2 , x=4 aThe bounded area is as shown in fig. Required area isA=2∫

a

4a

2√axdx

A=4 √a [ 2 x√ x3 ]4 aaA=8√a

3(4a√4 a−a√a)

A=56 a2

3sq .units

50 (c)

A=∫0

2

ex dx

A=[ex ]20

A=e2−1 sq .units

51 (c)Curve areay= log xx=1 and x=eRequired Area isA=∫

1

e

log xdx

A=¿

A=(e−0 )−[x ]e1

A=e−e+1A=log e sq. units

52 (c)The bounded area is as shown in figEqution of circlex2+ y2=r2Required area isA=4∫

0

r

√r2−x2dx

P a g e |26

A=4 [ x2 √r2−x2+ r2

2sin−1( xa )] ro

A=4 [ r2 √r2−r2+r2

2sin−1 (1 )−0−r 2

2sin−10]r0

A=2r 2( π2 )A=π r2 sq .units

53 (d)

Area=4∫0

3

√9−x2dx

Area=4 { x3 √9−x2+ 92sin−1( x3 )}30

Area=4 {92 ( π2 )}Area=9π sq .units

54 (d)

Area=4∫0

4

√16−x2dx

Area=4 { x4 √16−x2+162sin−1( x4 )}40

Area=4 {8( π2 )}Area=16π sq .units

55 (b)The bounded area is shown in fig.

The curve isx2+ y2=2ax(x−a)2+ y2=a2 represented a circle with center at (a ,0 ),And radius aRequired area isA=∫

0

a

√2ax−x2dxA=∫

0

a

√a2−( x−a )2dx

A= x−a2 [ x2 √a2− (x−a )2+ a

2

2sin−1( x−a

a )]a0A=0+ a

2

2sin−10−a2

2sin−1(−1)

A=π a2

4sq .units

56 (a)The bounded area is shown in fig.Required area isArea = 4 Area OABA=4∫

0

a ba √a2−x2dx

A=4ab [ x2 √a2−x2+ a

2

2sin−1( xa )]a0

A=4ab

[0+ a2

2sin−1 (1 )−0−a2

2sin−1(0)]

A=2ab( π2 )A=πab sq .units

P a g e |27

57 (c)

Area=4∫0

3 23 √9−x2dx

Area=83∫0

3

√32−x2dx

Area=83 {x2 √32−x2+ 9

2sin−1( x3 )}30

Area=83 ( 92 π2 )

Area=6π sq .units

P a g e |28

58 (d)

Area=4∫0

4 34 √32−x2dx

Area=4∫0

4

√42−x2dx

Area=3 {x2 √42−x2+ 162sin−1( x4 )}40

Area=3(8 π2 )Area=12π sq .units

59 (d)

Area=4∫0

5 45 √25−x2dx

Area=165 ∫

0

5

√52−x2dx

Area=165 {x2 √252−x2+25

2sin−1( x5 )}50

Area=165 ( 252 π

2 )Area=20π sq .units

60 (b)The bounded area is as shown in fig.Area of ellipse isπab sq.unitsArea of ellipse in firset quadrant is πab4 sq.unitsArea of OAB=12 ab sq.units

Required area isA=Area of ellipse in first quadrant – Area of OABA=πab

4−ab2

A=ab(π−2)

4sq .units

61 (c)The bounded area is as shown in fig.Required area isArea=2 AreaOCAD

Area=2∫0

ae ba √a2−x2dx

Area=2ba [ x2 √a2−x2+ a

2

2sin−1 x]ae0

Area=2ba ( ae2 √a2−a2e2+ a

2

2sin−1( aea )−0)

Area=ba(a2 e√1−e2+a2 sin−1 e)

Area=ab(e√1−e2+sin−1 e)

62 (a)

y=x∧ y2=4 ax∴ x2=4ax∴ x ( x−4 a )=0∴ x=0 , x=4a

Area=∫0

4a

2√a√ xdx−∫0

4a

xdx

P a g e |29

Area=2√a [ x32

32 ] 4 a

0−[ x22 ]4 a0

Area=323a2−16

2a2

Area=8 a2

3sq .units

63 (b)

Area=∫0

4a

2√ xdx−∫0

4

xdx

Area=43

[ x 32 ] 4

0−[ x22 ]40

Area=323

−162

Area=83sq .units

64 (d)

Area=∫0

1

2√x dx−∫0

1

2xdx

Area=2( 23 )[ x32 ] 10−2[ x

2

2 ]10Area=4

3−1

Area=13sq .units

65 (d)The bounded area is as shown in figCurves arey=x2 and y=xSolvin themA≡(1,1)Required area isA=∫

0

1

(x−x2)dx

A=[ x22 −x3

3 ]11A=1

2−13

A=16sq .units

66 (d)The bounded area is as shown in figCurves arey2=x and 2¿ x+ ySolvin themA≡(1,1) and B≡(4 ,−2)Required area isA=∫

−2

1

(2− y− y2)dy

A=[2 y− y2

2−y3

3 ] 1−2A=2−1

2−13+4+2−8

3

P a g e |30

A=92sq .units 67 (d)

Area=18∫−1

8

1

( y+14 − y2

2 )dy

Area=18∫−1

2

1

(2+2 y−4 y2 )dy

Area=18 {2 y+ y2− 43 y3}

1−12

Area=18 ( 53 + 7

12 )Area= 9

32sq .units

P a g e |31

68 (b)The bounded area is as shown in figCurves arey2=4 x and 3y¿2 x+4Solvin themy2=2 (3 y−4 )y2−6 y+8=0( y−2 ) ( y−4 )=0y=2, y=4x=1,4A≡(1,2) and B≡(4,4)Required area isA=∫

1

4

(√4 x−( 2 x+43 ))dxA=[2( 2x √ x

3 )−13 (x2+4 x )] 41A=1

3[32−(16+16 )−4+5]

A=13sq .units

69 (d)The bounded area is as shown in figCurves arey2=x and 2¿ x+ ySolvin themA≡(1,1) and B≡(1,0) and C≡(2,0)Required area isA=¿Area OAB +Area ABCA=∫

0

1

√x dx+∫1

2

(2−x)dx

A=[ 2x √x3 ]

1

0+[2 x− x2

2 ]21

A=23+4−2−2+ 1

2

A=76sq .units

70 (d)The bounded area is as shown in figCurves arey2=16 xEquation of chord AB isy−412−4

= x−99−1

y−48

= x−98

x− y+3=0Required area isA=∫

1

9

(4 √x−( x+3 ) )dx

A=[ 8x √ x3

−x2

2−3x ]91

A=72−812

−27−83+ 12+3

A=163sq .units

71 (a)The bounded area is as shown in figCurves are4 y=3 x2 and 2y¿3 x+12Solvin them6 x+24=3 x2

P a g e |32

x2−2 x−8=0x=4 ,−2y=12,3Required Area isA=∫

−2

4

( 3 x+122−3 x

2

4 )dxA=[ 12 (3 x22 +12x )− x3

3 ] 4−2A=3

4[16−4 ]+6 [4+2 ]−1

4[64+8]

A=9+36−18A=3 sq .units

72 (c)The bounded area is as shown in figCurves arey=x2 and y¿ x+2Solvin themx2=x+2x2−x−2=0x=−1 , x=2y=1 , y=4A≡(−1,1) and B≡(2,4 )Required area isA=∫

−1

2

(x+2−x2)dx

A=[ x22 +2x− x3

3 ] 2−1A=2+4−8

3−12+2−1

3

A=92sq .units

73 (b)The bounded area is as shown in figCurves arey=x2+1 is parabola vertex at (2,0)y=2 ( x−1 ) is straight lineSolving them4 ( x−1 )2=16 ( x−2 )x2+2 x−1=4 x−8x2−6 x+9=0( x−3 )2=0x=3 , y=4A≡(1,0) and B≡(3,4 ) and C≡(2,0) and D≡(2,0)Required area isA=¿Area ABD - Area BCDA=∫

1

3

y linedx−∫2

3

y paraboladx

A=∫1

3

2 ( x−1 )dx−∫2

3

4√ x−2dx

A=[ x2−2x ] 31−4 [ 2 ( x−2 )

32

3 ]32A=9−6−1+2−8

3(1)

A=43sq .units

P a g e |33

74 (a)

Area=∫0

2

(x2+1−x )dx

Area=[ x33 +x− x2

2 ]20Area=8

3+2−2

Area=83sq .units

P a g e |34

75 (c)The bounded area is as shown in figCurves arex2+2= y and x= ySolvin themA≡(3,3) and B≡(3,11) and C≡(0,2)Required area isA=∫

0

3

(x2+2−x )dx

A=[ x33 +2x− x2

2 ]30A=9+6−9

2−0

A=212sq .units

76 (c)

Area=∫0

1

[ x+1−(x2−1)]

Area=∫0

1

(x−x2)dx

Area=[ x22 −x3

3 ]10Area=1

2−13

Area=16sq .units

77 (c)The bounded area is as shown in fig

Curves arey=x2−6 x+11y¿ x2−6 x+9+2

y−2=( x−3 )2 represented parabola with vertex at (3,2)y=5−x is straight lineSolving themx2−6 x+11=5−xx2−5 x+6=0( x−2 ) ( x−3 )=0A≡(2,3) ,B≡(3,2)Required area isA=∫

2

3

((5−x)−(x2−6 x+11) )dx

A=[ 5x22 −x3

3−6 x ]32

A=452

−9−18−10+ 83+12

A=16sq .units

78 (d)

Area=∫0

4a

(2√a√x− x2

4 a )dxArea=2√a ( 23 )[ x

32 ] 4 a

0− 14 a

[ x33 ]4 a0Area=4

3 √a (√4a )3− 112a

(4 a )3

Area=323a2−16

3a2

Area=16 a2

3sq .units

P a g e |35

79 (c)

Area=∫0

1

√x dx−∫0

1

x2dx

Area=23

[ x32 ] 1

0− 13

[ x3 ] 10

Area=23−13

Area=13sq .units

80 (d)

Area=∫0

4

(2√ x− x2

4 )dxArea=2( 23 )[ x

32 ] 4

0− 112

[ x3]40

Area=43

(8 )−6412

Area=163sq .units

81 (d)

Area=∫0

12

¿¿ )dxArea=√12( 23 )[ x

32 ] 12

0− 136

[ x3 ]120

Area=23

(144 )−13(144)

Area=1443

Area=48 sq .units

82 (a)

Parabolaare y2=9 x4

∧x2=16 y3

4 a=94,4b=16

3

a= 916, b=4

3Required area isA=16

3 ( 916 )( 43 )A=4 sq. units

83 (c)The bounded area is as shown in figCurves arey2=x+1 and y2=−x+1Solvin themA≡ (−1,0 ) and B≡ (1,0 ) and C≡ (0,1 ),D≡(0 ,−1)Required area isA=2 ( Area I )+2(Area II )

P a g e |36

A=2∫−1

0

√x+1dx+2∫0

1

√−x+1dx

A=2[2 ( x+1 )32

3 ] 0−1+2[2 ( x−1 )

32

−3 ]10A=4

3[1+0+1]

A=83sq .units

84 (d)Area=¿

Area=∫0

4

xdx+∫4

4√2

√32−x2dx

Area=[ x24 ]4

0+[ x2 √32−x2+ 322sin−1( x

4√2 )]4 √24

Area=8+16sin−1 (1 )−[8+16sin−1( 1√2 )]Area=8+16( π2 )−8−16 ( π4 )Area=4 π sq .units

85 (b)The bounded area is as shown in fig

Curves arex2+ y2=36 and x+ y=6Required area isA=Area of the ¿̊ the first quadrant−Areaof ∆OAB

A=π4

(6 )2−12

(6 )(6)

A=9π−18sq .units

86 (a)The bounded area is shown as followEllipse is x2

9+ y

2

4=1Line is2 x+3 y=6From fig.A≡(3,0),B≡(0,2)Required area is

A=∫0

3

( 23 √9−x2−23

(3−x ))dxA=2

3 [ x2 √9−x2+ 92sin−1( x3 )−3 x+ x

2

2 ]30A=2

3 ( 92 sin−1 (1 )−9+ 92 )

A=32

(π−1 ) sq .units

87 (b)The bounded area is shown as followP a g e |37

The curves arey=x2+5 and y=x3Required area is

A=∫1

2

(x2+5−x3)

A=[ x33 +5 x− x4

4 ]21A=8

3+10−16

4−13−5+ 1

4

A=4312

sq .units

88 (c)The bounded area is shown as followThe curves arey=2x ,2 y=x and 4=xSolving themA≡(4,8) ,B≡(4,2)Required area isA=∫

0

4

(2 x− x2 )dx

A=[ x2− x2

4 ]40A=3

4(16)

A=12 sq .units

89 (c)The bounded area is shown as followThe curves are

y=2x+1 , y=3x+1 and 4=xSolving themA≡(0,1) ,B≡ (4,13 ) , C≡(4,9), D≡(4,0)Required area isA=Area ABCA=¿Area OBAD – Area OACDA=∫

0

4

(3x+1 )dx−∫0

4

(2x+1)dx

A=[ 3x22 +x−x2−x] 40A=[ x22 ]40A=8 sq .units

90 (b)A=A (∆ ABC )

A=∫1

3

( x+92 +2x−7)dx+∫3

5

( x+92 −3 x+8)dxA=1

2∫13

(5 x−5)dx+ 52∫35

(5−x )dx

A=12 [ 5 x22 −5x ] 3

1+ 52[5 x− x2

2 ]53A=1

2 ( 452 −15−52+5)+ 52 (25−252 −15+ 9

2 )A=10 sq .units

91 (d)A=A (∆ ABC )

P a g e |38

A=∫2

3

[ (3 x−5 )−( x+13 )]dx+∫35

[ (−x+7 )−( x+13 )]dxA=1

3∫23

(8 x−16)dx+13∫3

5

(−4 x+20)dx

A=83 [ x22 −2 x ] 3

2−43[ x22 −5 x ]53

A=83 [ 92−4]−43 [ 162 −10]

A=123

A=4 sq . units

92 (c)A=A (∆ ABC )

A=12 |2 0 14 5 16 3 1|

A=12|4−18|

A=12|−14|

A=7 sq .units93 (a)The bounded area is shown as followThe curves are

y=sin x and y=cos xRequired area isA=∫

0

π4

sin xdx+∫π4

π2

cos xdx

A=[−cos x ]π4

0+ [sin x ]

π2π4

A=−1√2

+1+1− 1√2

A=2−√2 sq .units

94 (c)The bounded area is shown as followThe curves arey=sin x and y=cos xSolving them

sin x=cos x

x=π4, y= 1

√2

A≡( π4 , 1√2 )Required area isA=∫

0

π4

(cos x−sin x)dx

A=[sin x+cos x ]π40

A= 1√2

+ 1√2

−0−1

A=√2−1 sq .units

95 (c)The area bounded by the curves as shown in fig.Required volume isV=π∫

0

2

x2dy

V=π∫0

2

y2dy

P a g e |39

V=π [ y33 ]20V=8 π

3cu .units

96 (a)

Volume=π∫0

4

y2dx

Volume=π∫0

4

x2dx

Volume=π3

[x3 ] 40

Volume=64π3

cu .units

97 (d)

Volume=π∫0

3

y2dx

Volume=π∫0

3

4 x2dx

Volume=43π [ x3 ]3

0Volume=36π cu .units

98 (b)

Volume=π∫0

2

( x+1 )2dx

Volume=π3 [ ( x+1 )3 ]2

0

Volume=26 π3

cu .units

99 (c)The area bounded by the curves as shown in fig.Curves are xy=2And y=1 to x=4Required volume isV=π∫

1

4 4y2dy

V=4 π [−1y ]41V=4 π [−14 +1]V=3π cu .units

P a g e |40

100 (a)

Volume=π∫1

2 1x2dx

Volume=π [−1x ]21Volume=π

2cu .units

101 (d)The area bounded by the curves as shown in fig.Curves are y=sin xRequired volume isV=π∫

0

π2

sin2 xdx

V=π∫0

π2

( 1−cos x2 )dxV= π

2 [x−sin 2 x2 ] π20

V= π2 ( π2−sin π

2 )V= π2

4cu .units

P a g e |41

102 (b)The area bounded by the curves as shown in fig.Curves is y2=4axLatus rectum is x=aRequired volume isV=π∫

0

a

4 axdx

V=4a π [ x22 ]a0V=2π a3 cu . units

103 (a)

Volume=π∫0

4

16 xdx

Volume=8 π [ x2 ] 40

Volume=128π cu .units

104 (d)

Volume=π∫0

2

16 xdx

Volume=8 π [ x2 ] 20


105 (a)

Volume=π∫0

4

(4 x−x2)dx

Volume=π {2 x2− x3

3 }40Volume=π (32−643 )Volume=32 π

3cu .units

106 (c)

Volume=π∫0

1

(4 x−4 x2)dx

Volume=4 π {x22 − x3

3 }10Volume=2 π

3cu .units

107 (b)

P a g e |42

Volume=π∫0

3

(3 x−x2)dx

Volume=π {3x22 − x3

3 }30Volume=π ( 273 −27

3 )Volume=9π

2cu .units

108 (d)

Volume=π∫0

1

(x2−x4 )dx

Volume=π { x33 − x5

5 }10Volume=π ( 215 )Volume=2 π

15cu .units

109 (c)

Volume=π∫0

2

(4 x2−x4 )dx

Volume=π {43 x3− x5

5 }20Volume=π ( 323 −32

5 )Volume=64π

15cu .units

110 (a)The area bounded by the curves as shown in fig.Curves is x2=4 yx=−3 , x=4Required volume isV=π∫

−3

4

y2dx

V=π∫−3

4

( x416 )dxV=

π16 [ x55 ] 4−3

V= π80

(1240+243 )

V=1267 π80

cu .units

111 (d)Equation of chord AB is y=x+3

Volume=π∫1

9

[− (x+3 )2+16 x ]dx

Volume=π [−( ( x+3 )3

3 )+8 x2]91Volume=π [ (−576+648 )−(−643 +8)]Volume=256 π

3cu .units

P a g e |43

112 (a)The area bounded by the curves as shown in fig.Curves is x2=4 y , y2=4 xSolving themA≡(4,4)Required volume isV=π∫

0

4

(4 x− x4

16 )dxV=π [2 x2− x5

80 ] 40V=π (32−102480 )V=32π ( 5−25 )V=96 π

5cu .units

P a g e |44

113 (d)

Volume=π∫0

1

(x−x4 )dx

Volume=π ( x22 − x5

5 )10Volume=3 π

15cu .units

114 (d)The area bounded by the curves as shown in fig.Curves isx2+ y2=25 and 3 x=4 ySolving themx=4 , y=3A≡ (4,4 ) ,B≡ (4,0 ) ,C≡(5,0)Required volume isV=π∫

0

4

y line2 dx+π∫

4

5

y¿̊2dx

¿

V=π∫0

4 9 x2

16dx+π∫

4

5

(25−x2)dx

V=9 π16 [ x33 ] 4

0+π [25 x− x3

3 ]54V=3 π

16(64 )+π (125−1253 −100+ 64

3 )V=36 π+14 π

3

V=50 π3

cu . units

115 (d)The area bounded by the curves as shown in fig.Curves isx2+ y2=36 and x+ y=6Solving themA≡ (6,0 ) , B≡ (0,6 )Required volume isV=Volume of sector OAB−Volume of ∆OAB

V=π∫0

6

(36−x2) dx−π∫0

6

(6−x )2dx

V=π∫0

6

(36−x2−36+12 x−x2)

V=π [6 x2− 2x33 ]60V=π (216−23 (216 ))V=72 π cu .units

116 (d)Curves isx2+ y2=36 and x=0 , x=3Required volume isV=π∫

0

3

y2

V=π∫0

3

(25−x2)dx

V=π [25 x− x3

3 ]30V=π (75−9)V=66π cu .units

117 (c)The area bounded by the curves as shown in fig.Curves isx2+ y2=4 and y2=3 xSolving them

P a g e |45

x=1 , y=√3A≡ (1 ,√3 ) ,B≡ (1,0 ) ,C≡(2,0)Required volume isV=π∫

0

1

y parabola2 dx+π∫

1

2

y¿̊2dx

¿

V=π∫0

1

3 xdx+π∫1

2

(4−x2)dx

V=π [3 x22 ] 10+π [4 x− x3

3 ]21V=π ( 32 +8−8

3−4+1

3 )V=π (4+ 32−73 )V=π

V=19 π6

cu .units

118 (d)The area bounded by the curves as shown in fig.Curves isx2

a2+ y

2

b2=1

Solving themA≡ (a ,0 ) , A ' ≡ (−a ,0 )Required volume isV=π∫

−a

a

y2dx

V=π∫−a

a

b2(1− x2

a2 )dxV= 2π b

2

a2∫0

a

(a2−x2)dx

V=2π b2

a2 [a2 x− x3

3 ]a0

V=2π b2

a2 (a3−a3

3 )V= 4π b

2a3

cu .units

119 (d)

Volume=π∫−4

4

9 (1− x2

16 )dxVolume=9π

16 ∫−44

(16−x2)dx

Volume=9π16 {16 x− x3

3 } 4−4Volume=9π

16 {(64−643 )−(−64+ 643 )}Volume=9π

16 (128−1283 )Volume=48 π cu .units

120 (d)

Volume=π∫−4

4

9 (1− x2

16 )dxVolume=18 π

16 ∫0

4

(16−x2)dx

Volume=9π8 {16 x− x3

3 }40Volume=9π

8 (64−643 )Volume=144 π

3cu .units

P a g e |46

121 (d)

Volume=π∫−3

3 49

(9− y2)dy

Volume=8π9 ∫

0

3

(9− y2)dy

Volume=8π9 (9 y− y3

3 )30Volume=24 π ( 23 )Volume=16π cu .units

122 (a)The area bounded by the curves as shown in fig.Curves isx2

a2+ y

2

b2=1

Major axis is AA’And Minor axis is BB’Solving themA≡ (a ,0 ) , B≡ (0 , b )Equation of chordxa+ yb=1

Required volume isV=π∫

0

a

¿¿¿)dxV=π∫

0

a

( b2a2 (a2−b2 )−b2

a2(a−x )2)

V= π b2

a2∫0

a

(a2−x2−a2+2ax−x2 )dx

V=π b2

a2 [ax2−2 x33 ]a0V= π b2

a2 (a3−2a33 )V= πab2

3cu . units

123 (c)The area bounded by the curves as shown in fig.Curves isx2− y2=a2

A≡ (a ,0 ) , B≡ (2a ,√3 a ) ,C≡(2a ,0)Required volume isV=π∫

a

2a

( x2−a2)dx

V=π [ x33 −a2x ]2aaV=π [8a33 −2a3−a3

3+a3]

V= 4π a3

3cu .units

124 (c)A sphere is generated by revolving the area ofSemi circle x2+ y2=r2 from x=−r ¿ x=rAs shown in figureRequired volume isP a g e |47

V=π∫−r

r

y2dx

V=π∫−r

r

(r 2−x2)dx

V=2π∫0

r

(r2−x2)dx

V=2π [r2 x− x3

3 ]r0V=2π (r3− r3

3 )V= 4π r

3

3cu .units

125 (a)

Volume=π∫−4

4

(16−x2)dx

Volume=2π∫0

4

(16−x2)dx

Volume=2π [16 x− x3

3 ]40Volume=2π (64−643 )Volume=2π ( 23 )Volume=256 π

3cu .units

126 (c)The area bounde by the curve is as shown

in figureA≡ (0 , r ) ,B≡ (h , r ) ,C≡(h ,0)Equation of AB is y=rRequired volume isV=π∫

0

h

y2dx

V=π∫0

h

r2dx

V=π r2 [x ]h0

V=π r2hcu .units

127 (d)The area bounde by the curve is as shown in figureA≡ (h , r ) ,B≡ (h ,0 ) ,Equation of OA isy= rx

hRequired volume isV=π∫

0

h r2x2

h2dx

V=π r2

h2 [ x33 ]hoV= π r2h

3cu .units

128 (d)

Area=∫−π

π

sin xdx

P a g e |48

Area=2∫0

π

sin x dx

Area=2 (−cos x )π0

Area=2(1+1)Area=4 sq .units

129 (b)

Area=∫0

4

y2dy

Area=13[ y3]4

0

Area=643sq .units

130 (a)

A=∫0

1

x2dx

A=13sq .units

131 (d)It is a square of diagoanal of length 4 units

A=(2√2 )2

A=8 sq .units


1

2 1xdx

A=[ log x ]21

A=( log2)sq .units


0

2

(2 x−x2)dx

A=[ x2− x3

3 ]20A=4−8

3

A=43sq .units

P a g e |49

134 (b)

∫0

16 /m2

(√16 x−mx )dx=23

x=16m2

[4× 23 x32−m x2

2 ]16/m2

0=23

¿> 83× 64m2

−m2256m4 =2

3¿>m=4


0

1

(√ x−x2)dx

A=[ 2x 323 − x3

3 ]10A=( 23−13 )A=1

3sq .units

P a g e |50

136 (b)Required area isA=πab

4−Areaof ∆OAB

A=5 (π−2 ) sq .units

137 (b)Required area isA=4 [ 12×1×1]A=2 sq .units

138 (b)Required area isA=∫

0

π /4

(cos x+sin x)dx

A=[sin x+cos x ] π /40

A=(√2−1 ) sq .units

139 (a)Required area isA=1

2(PQ )(AL)

A=12

(1 )(1)

A=12

[∴BC=2 , PQ=12BC=1 ,LA=1]

140 (a)The curve y=x3∧ y=√ x… ( i )( y ≥0)Points of intersection of curve (i) and y=x3…(ii)Are (0,0)(1,1)Required area isA=∫

0

1

( y1− y2)dx

A=∫0

1

(√ x¿−x3)dx¿

A=( 2 x32

3− x4

4 )10A=2

3− 14

A=8−312

A= 512

141 (a)Given function is y=|x−1|i.e.y=x−1x>1¿−x+1 x<1Required area is

A=12×1×1

P a g e |51

A=1

142 (b)Required area isA=∫

0

3

xdy

A=∫0

3 y2

4dy

A=[ y312 ]30A=27

12

A=94sq .units

143 (c)

Areaof ∆ AOB=12×1× 1

2

Areaof ∆ AOB=14

A=14sq .units

144 (c)

Areaofrhombus=12× (Product of diagonals )

Areaofrhombus=122ca2cb

Areaofrhombus=2c2

absq .units

145 (d)End points of latus rectum in 1st quadrant is (ae , b2a )Equation of tangent at (ae , b2a ) i.e. (3e , 253 )is

It intersects X-axis at ( 3e ,0) and y-axis at (0,3)

Areaof ∆OAB=12 ( 3e ).3= 9

2e

So, Area of quadrilateral ABCD=4. 92eAlso,b2=a2(1−e2)e2=1−b2

a2=1−5

9=49

¿>e=23

146 (c)We have,y=log x , x=1 , x=2Required area isA=∫

1

2

ydx

A=∫1

2

log xdx

P a g e |52

A=[x log x−x ]21

A=2 log2−2− log1+1A=2 log2−1A=2 log2−log eA=log 4− loge

A=log 4esq .units

147 (d)Required area isA=A (∆OAB )+A (∆OCD)

A=12×1×1+1

2×2×2

A=52sq .units

148 (a)Equation of parabola isy=x2…(i)And equation of the straight line isy=x… ( ii )

From (i) and (ii) ,we getx2−x=0¿>x=0∨x=1¿> y=0 or y=1Hence,the co-ordinates of their points of intersction are O(0,0) and P(1,1)∴ Required area between parabola and straight lineA=∫

0

1

xdx−∫0

1

x2dx

A=[ x22 −x3

3 ]10

A=[ 12− 13 ]A=1

6sq .units

149 (a)Required area isA=A (∆ ABD )+A(∆ ACE )

A=12×1×1+1

2×1×1

A=12+12

A=1 sq .units150 (a)Required area is

A=∫0

9

√x dx−∫3

9

( x−32 )dx

A=[ x32

32 ] 9

0−12[ x22 −3 x ]93

A=( 23 ,27)−12 {( 812 −27)−( 92−9)}A=18−9A=9 sq .units

151 (a)Required area isA= ∫

−π /2

π /2

ydx

A=2∫0

π /2

sin xdx

P a g e |53

A=2 [−cos x ]π /20

A=2 sq .units

152 (c)

x2

a2+ y

2

b2=1 , a>b ,

About major axisy2=b2(1− x2

a2 )y2=b2

a2(a2−x2 )

Volume=2π∫0

a

y2dx

Volume=2[π∫0a b2

a2(a2−x2 )dx ]

Volume=2 π b2

a2 (a2 x− x3

3 )a0Volume=2 π b

2

a2 (a3−a3

3 )Volume=2 π b

2

a2× 2a

3

3

Volume=43π ab2

153 (b)Required volume isV=π∫

0

2

x2dy

V=π∫0

2 y4dy

V=π4 [ y22 ] 20

V= π8

(16−0)

V=2π cu .units154 (b)Required volume is

V=π∫0

16

x2dy

V=π∫0

16 y4dy

V= π8

[ y2 ] 160

V=256 π8

V=32π cu .units155 (c)

the parabola x2=4 yi.e. y= 14 x2 passes throughthe points (0,0)(4,4) and (-4,4) and its axis of symmetry is X-axis.Now the volume generated by revolving the area bounded by the curvey= 14 x2,Y-axis y=0 and the line x=−3 , x=4 about X-axisV=π∫

−3

4

( 14 x2)2

dx

V= π16∫−3

4

x4dx

V=π16 [ x55 ] 4−3

V= π80 [ (4 )5−(−3 )5 ]

V= π80

[1024+243 ]

V= π80

[1267]

V=15.837 π c .u .156 (b)The volume of the solid generated by revolving the region bounded by the curve x= 2y ,Y-axis x=0The line y=1∧ y=4 about Y-axis is

P a g e |54

V=π∫1

4

[ f (x ) ]dy

V=π∫1

4

( 2y )2

dy

V=4 π∫1

4 dyy2

V=4 π [−1y ]41V=4 π [−14 +1]V=3π cu . units

157 (a)

Volumeof cone=13π ×16×4

Volumeof cone=64 π3

158 (b)

Required area isA=∫

−1

1

xdy

A=∫−1

1

y3dy

A=2∫0

1

y3dy

A=( 2 y44 )10A=1

2159 (a)The triangle area is

A=∫0

2

xdy

A=∫0

2 12ydy

A=12 [ y22 ]20

A=14[4−0 ]

A=1 sq .units

160 (c)Required area isA=∫

1

4

y . dx

A=∫1

4

4 √x .dx

P a g e |55

A=4 [ x32

32 ]41

A=83[4

32−13 /2]

A=83[8−1]

A=83×7

A=563sq .units

161 (b)Let LL’ be the latus rectum and S(1,0) be the focusOf the parabola y2=4ax∴Eqof letusrectum is x=1∴Required area=2× Areaof regionOSLO

A=2∫0

1

ydx

A=2∫0

1

2√ xdx

A=4∫0

1

x32 dx

A=4.[ x32

32 ]10

A=83

[132−0]A=8

3sq .units

162 (a)Required area=Area of regionOABO+¿Area of region BCDBA=∫

0

π

sin xdx+|∫π

2π

sin xdx|

A=[−cos x ]π

0+|[−cos x ]2ππ |

A=−cos π+cos0+|−cos2 π+cosπ|A=−(−1 )+1+|−1−1|A=2+2A=4 sq. units

163 (c)Equation of parabola is y=x2+2The line y=x passes through origin

Required area =Area under the parabola – Area under the lineA=∫

0

3

(x2+2 )dx−∫0

3

xdx

A=[ x33 +2x ] 3

0−( x2

3 )30

A=(9+6 )−( 92 )A=15−9

5

A=212

A=10.5 sq .units164 (d)The given curves are y2=x+1∧¿

P a g e |56

y2=−x+1These are two parabola’s whose vertices are (-1,0)And (1,0) respectively

Required Area isA=2[∫

−1

0

√x+1dx+∫0

1

√1−xdx ]

A=2{[ ( x+1 )32

32 ]

0

−1+[ (1−x )32

32 ]10}

A=43 { [1−0 ]+ [0+1 ] }

A=43×2

A=83sq .units

165 (b)The area bounded by the parabolax=f ( y )=1

2 √ y

Required Area isA=1

2∫14

xdy

A=12∫14

√ y dy

A=12∫14

y12 dy

A=12 [ y

32

32 ]41

A=13

[4 32−132 ]A= 4

3[8−1]

A=73sq .units

166 (c)Required Area isA=∫

a

b

log x .dx

A=[x . log x ]b

a−∫a

b

x . 1x dx

A=[x . log x−x ]ba

A=(b . log b−b ) .∫a

b

x . 1xdx

A=b ( log b−1 )−a (log a−1)A=b (log b−log e )−a (log a−log e)

A=−a log( be )+b log( ae )167 (a)The required area is shown by shaded portion

Required Area = 2× Area of region OABA=2×∫

0

π2

ydx

A=2∫0

π2

cos xdx

A=2 [sin x ]π20

P a g e |57

A=2(sin π2−sin 0)A=2(1−0)A=2 sq .units

168 (a)The area enclosed by the curves is shown in shaded region

At the point of intersection ofx2+ y2=32∧ y=x ,wehavex2+ y2=32=¿2 x2=32=¿ x2=16x=4Required Area =Areaof ∆OPQ+Area of ∆ PQA

A=∫0

4

xdx+∫4

4√2

√18−x2dx

A=[ x22 ]4

0+[ x2 √32−x2+ 322sin−1( x

4√2 )]4 √24

A=8+[0+16sin−11−8−16sin−1 1√2 ]A=16( π2 )−16 ( π4 )A=8π−4 πA=4 π sq .units

P a g e |58

169 (c)Equation of parabola is y2=16 x∴Focus=(4,0 )∧L .R . (¿' ) is x=4∴Required volumeis

V=∫0

4

π . y2dx

V=∫0

4

π .16 x .dx

V=16∫0

4

xdx

V=16 π [ x22 ]40V=16 π [8−0 ]V=128 π sq .units

170 (a)The given equation hyperbola isx2− y2=9=¿a2=9=¿a=3

Let S and S’ be its foci, A and A’ are the vertices of hyperbola the∴Required volume (V )=2×volume obtainedby revolving aboutX−axis the area ALS

∴Eccientricity of the hyperbola=√a2+b2a

=√2

∴Focus S=(3√2 ,0)

V=2π∫3

3√2

y2dx

V=2π∫3

3√2

(x2−9)dx

V=2π [ x33 −9 x ]3√23V=2π [36 √2

2−27√2−9+27]

V=2π [18√2−27√2+18 ]V=2π [18−9√2 ]V=18 π (2−√2 ) c .u

171 (b)The straight line y=2x passes through the originAnd P≡(0,6) Volume of the required region

V=π∫0

3

(2 x )2dx

V=4 π∫0

3

x2dx

V=4 π [ x33 ]30V= 4π

3[33−0]

V=36π cu .units172 (c)The curvey2=4axis symmetric about X-axis .Its vertex at origin and focus S≡ (a ,0 )Ends of latus rectum are (a ,2a )∧(a ,−2a )Required volume is

V=2π∫0

a

y2dx

V=2π∫0

a

4 axdx

V=8πa [ x22 ]a0V=8πa [ a22 −0 ]V=4 π a3 cub .units

173 (b)The lines y=2x+1 , y=3x+1 intersects at A(0,1).The linesy=3 x+1∧x=4 intersects P a g e |59

at B(4,13).The lines y=2x+1and x=4 intersects at C(4,9)

∴Therequired areaisA=Area (OABCD )−Area (OACD )

A=∫0

4

[ f ( x )−g( x)]dx

Where f (x )=3x+1 , g ( x )=2 x+1

A=∫0

4

[ (3 x+1 )−(2 x+1)]dx

A=∫0

4

xdx

A=[ x22 ]40A=16

2A=8 sq .units

174 (c)The sphere is the solid of revolution generated by the revolution of semi-circular area about its diameter

Equation of circle x2+ y2=a for the semi circle about X-axis the variable varies from x=−a ,x=aVolumeof sphere is

V=π∫−a

a

y2dx

V=π∫−a

a

(a2−x2 )dx

V=2π∫0

a

(a2−x2)dx

V=2π [a2 x− x3

3 ]a0V=2π [a3− a3

3 ]V= 4

3π a3cu .units

175 (b)

The curve is y=xRequired area = Area OCD + Area OABA=∫

−1

0

− ydx+∫0

2

ydx

A=∫−1

0

−xdx+∫0

2

xdx

A=−12

[x2 ]0

−1+12

[x2 ] 20

A=−12

(0−1 )+ 12(4−0)

A=52

176 (a)

The curve is x2a2

+ y2

b2=1=¿ y=b

a √a2−x2

Required area A = 4.Area OABOA=4∫

0

π /2 ba √a2−x2dx

Put x=a sinθ=¿dx=acosθ .dθ

P a g e |60

A=4∫0

π /2

a .b cosθ

A=4 a .b 12. π2

A=πab

177 (d)

The required volume isV=π∫

0

2

y2dx

V=π∫0

2

( x+1 )2dx

V=π∫0

2

(x2+2 x+1)dx

V=π [ x33 +x2+ x]20V=26 π

3

P a g e |61

178 (c)The equation of circle x=r .cosθ , y=r . sinθ∴ Required surface areaS=∫

0

π

2πy √( dxdθ )2

+( dydθ )2

dθ

S=∫r

π

2πr sinθ √r2+sin 2θ+r2+cos2θ dθ

S=2π r2∫0

π

sin θ .dθ

S=2π r2 (−cosθ )π1

S=2π r2(1+1)S=4 π r2

179 (c)

Volume of revolution isV=∫

2

4

π y2dx

V=∫2

4

π x4dx

V=[ π x55 ]42V= π

5(45−25 )

V= π5

(1024−32)

V=9923

π cu .units

180 (c)

We have,y2=4 x vertex =(0,0)The equation of latus rectum isGiven by x=a i.e.,x=1V=∫

0

1

π y2dx

V=∫0

1

π .4 x .dx

V=4 x [ x22 ]10V=2π

181 (c)

Area=∫1

e

log xdx

Area=∫1

e

1. log xdx

Area=[x log x−x ] e1

Area=1 sq .units182 (c)

Area=∫1

4

2√x dx

Area=2( 23 )[ x32 ]41

Area=43(8−1)

Area=283sq .units

183 (c)

Area=∫0

1

[ ( x+1 )−(x2+1 ) ]dx

Area=∫0

1

(x−x2)dx

Area=[ x22 −x3

3 ]10Area=1

2−13

P a g e |62

Area=16sq .units

184 (b)

Area=2∫0

a

2√a√ xdx

Area=4 √a ( 23 )[x32 ]a0

Area=83 √a(a√a)

Area=8 a2

3sq .units

185 (c)

Area=∫0

4

[ (3x+1 )− (2 x+1 ) ] dx

Area=[ x22 ]40Area=16

2Area=8 sq .units

186 (d)

Area=∫0

3

(x2+3)dx

Area=[ x33 +3 x]30Area=( 273 +9)−0Area=18 sq .units

187 (a)

Area=|∫−1

0

xdx|+∫02

xdx

Area=|[ x22 ] 0−1|+( x22 )20Area=1

2+2

Area=52sq .units

188 (b)Required area =Area of ellipse in first quadrant –A(∆ AOB)

Area=∫0

a ba √a2−x2dx−1

2ab

Area=ba {x2 √a2−x2+ a

2sin−1( xa )} a

0−12ab

Area=ba ( a22 . π2 )−12 ab

P a g e |63

Area=14πab−1

2ab

Area=ab(π−2)4

sq .units

189 (a)Required area isArea= ∫

0

4a /m2

2√a−x2dx− ∫0

4a /m2

mxdx

Area=2√a ( 23 )[ x32 ]4a /m2

0−m2

[ x2 ]4 a/m2

0

Area=4√a3 ( 8a√a

m3 )−m2 ( 16a2m4 )

Area= 8a2

3m3 sq .units

190 (d)Given curve isy2=4axAt t=1 , x=at=2 , x=4 a

Area=2∫a

4a

2√a√x dx

Area=4 √a ( 23 )[x32 ]4aa

Area=83 √a (8a√a−a√a )

Area=56 a2

3sq .units

191 (a)

Area=∫π /4

π /2

sin2 xdx+ ∫π /2

3π /4

sin 2 xdx

Area=[−cos2 x2 ]π /2

π /4+[−cos2x2 ]3π /4π /2

Area=−12

(−1−1)

Area=1 sq .units

192 (b)

Area=∫0

c

√kcdx

Area=2√k3

[ (√x )3 ]c0

Area=2√kc3

(c)

Area=2c3 √kc sq .units

193 (c)

Area=∫1

3

x3dx

Area=14

[x 4 ]31

P a g e |64

Area=14(80)

Area=20 sq .units

194 (a)

Area=∫0

π /6

cos3 xdx

Area=13

[sin3 x ] π /60

Area=13(1)

Area=13sq .units

195 (c)

Area=4∫0

r

√r2− x2dx

Area=4 [ x2 √r2−x2+ r2

2sin−1( xr )] r0

Area=4 ( r22 )( π2 )Area=π r2 sq .units

196 (c)

Volume=π∫0

2

x2dy

Volume=π∫0

2

y2dy

Volume=π3

[ y3 ]20

Volume=8π3cu .units

197 (c)

Volume=π∫0

4

y2dx

Volume=π∫0

4

r 2dx

Volume=πr 2 ( x ) 40

Volume=πr 2hcu .units

198 (d)

Volume=π∫0

h r2 x2

h2dx

Volume=π r2

3h2(x3)h

0

Volume=π r2h3

cu .unitsl

P a g e |65

199 (d)

Volume=π∫0

3

4 x2dx

Volume=4 π3

(x3)30


P a g e |66

200 (a)

Volume=π∫−3

4 x4

16dx

Volume= π80

[ x5 ] 4−3

Volume= π80

(1024+243)

Volume=1267 π80

cu .units

201 (d)

Volume=π∫0

3

(25−x2)dx

Volume=π3

(75x−x3 )30

Volume=π3

(225−27)

Volume=66 π cu .units

202 (c)

Volume=π∫a

2a

( x2−a2 )dx

Volume=π [ x33 −a2 x ]2aaVolume=π ( 2a33 + 2a

3

3 )Volume=4 π a

3

3cu .units

203 (d)

Volume=π∫0

π2

sin 2 xdx

Volume=π2 {∫

0

π2

1.dx−∫0

π2

cos2 xdx}Volume=π

2 {[ x ]

π2

0−[ sin 2 x2 ]π20 }

Volume=π2 {π2−0}

Volume=π2

4sq .units

204 (c)

Volume=π∫0

1

3 x .dx+π∫1

2

(4−x2)dx

Volume=32π [ x2 ] 1

0+π [4 x− x3

3 ]21Volume=π {32+8−83−4+ 13 }Volume=19 π

6cu .units

P a g e |67

205 (d)

Volume=π∫1

4 1x2dx

Volume=π [−1x ]41

Volume=π (−14 +1)Volume=3 π

4cu .units

P a g e |68

Applications of Definite Integral

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