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Transcript of Homework Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd) Rogawski...
![Page 1: Homework Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f68b/html5/thumbnails/1.jpg)
Homework
Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
![Page 2: Homework Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f68b/html5/thumbnails/2.jpg)
Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
23. -axisx
2 22 2
0
2 4 2
0
25 3
0
6 2
36 4 4
32 32 32 4 64 0
5 3 5 3
960 96 160 704 704
15 15 15
V x dx
x x dx
x xx
V
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Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
25. 2y
22 2 2
0
2 4
0
25
0
2 6 2 2
16
32 16 32 0
5 5
160 32 128 128
5 5 5
V x dx
x dx
xx
V
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Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
27. 3x
2 2
26 2
2
6
2
63
2 2
2
2 2 2
2 3 0 3
2 3 2 9 9
2 2 3
322
18 12 2 8 2 4 0 24 24
y x y x x y
V y dy
y y dy
yyy
V
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Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
29. -axisx
2 222 4 2
0 0
25 3
0
2 4 4
4 45 3
32 32 8 0
5 3
96 160 120 376 376
15 15 15
V x dx x x dx
x xx
V
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Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
31. 6y
22 22 2 4 2
0 0
25 3
0
6 0 6 2 36 8 16
32 64 20 8 40 0
5 3 5 3
600 96 320 824 824
15 15 15
V x dx x x dx
x xx
V
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Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
33. 2x
2 2
262
2
63
226
2
2
2 2 2
2 2 2 2
2 8 4 4 2 2 8 2 4
3 22
8 8 8 12 8 18 4 0 2
3 3
72 64 32 32 8
3 3 3
y x x y x y
V y dy
y yy y dy y
V
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
235. , 12 , 0, about 2y x y x x y
2 2
3 22 2
0
3 2 4 2
0
35
3 2 4 2 3
00
12 12 0 4 3 0 4,3
12 2 2
196 28 4 4
192 28 3 192 145
243 2880 630 135 243 576 126 27
5 5
1872 1872
5
x x x x x x x
V x x dx
x x x x dx
xx x x dx x x x
V
5
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
37. 16 , 3 12, 0, about -axisy x y x x y
2
15 16 2
12 15
16 3 12 4 4 1 15
4 163
x x x x y
yV dy y dy
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Homework, Page 398
215 16 2
12 15
215 16 2
12 15
15 163 3
2 2
12 15
37. Continued.
4 163
8 16 256 32
9 3
4 16 256 16
27 3 3
3375 1728 300 240 192 192
27 27
yV dy y dy
yy dy y y dy
y yy y y y
4096 3375 4096 4096 3840 3600
3 3
3375 1620 1728 4096 720 3375 27 9 4
27 3 27 3
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
22
939. , 10 , about -axisy y x x
x
2 4 22
23 322 2 4 4
21 1
910 10 9 0 1, 3
910 100 20 81
x x x xx
V x dx x x x dxx
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Homework, Page 398
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2
3 322 2 4 421 1
3 33 5 3 3 5
3
1 1
39. Continued.
910 100 20 81
20 27 100 20 81 100
3 5 3 3 5
243 20 1 300 180 1 100 27
5 3 5
600 243 5 1500 1
5
V x dx x x x dxx
x x x x xx x
x
00 3 405 2544 1808 736
15 15 15
1472Two regions enclosed by the curves, means two volumes
15V
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
1 541. , , about -axis
2y y x y
x
2
22 22 2 2
0.5 0.5
22 3
12
1 52 5 2 0 0.5,2
2
12.5 6.25 5
25 5 1
4 2 3
25 8 1 25 5 1 10 2
2 3 2 8 8 24
75 60 16 3 75 15 1 48 136 109
6 24 24
y y y yy
V y dy y y y dyy
y yy
y
9
8
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
13 343. , , about -axisy x y x y
1 13 3 93 3
21 11 223 63 3
1 1
15 73
1
, 1,0 1
3 13 1 1
5 7 5 7 5 73
21 5 21 5 32 32
35 35 35 35
x y x y y y y y y
V y y dy y y dy
y y
V
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
45. , 1 , 0, about 4x xy e y e x y
2 2
0
1 2 1 0.5 0.69314718
4 1 4 2.148 6.748
6.748
x x x x
A x x
e e e e x A
V e dx
V
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Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
247. 4 , , 0, about -axisy x y x y x
2 2
4324 42 2 2
0 00
2 2 4 4 0 0,4
2 4 23
64 32 32 32
3 3 3
y x x x x x x x x
xV x x dx x x dx x
V
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Homework, Page 39849. Sketch the hypocycloid x 2/3 + y 2/3 = 1 and find the volume of the solid obtained by revolving it about the x-axis.
3 32 2 2 223 3 3 3
23 31 42 2
3 3
1 0
1 1 1
1 1
0.305 0.957 0.957
y x y x y x
V x dx x dx
V
y = 1/(x+0.62)-0.62
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Homework, Page 39851. A bead is formed by removing a cylinder of radius r from the center of a sphere of radius R. (Figure 12) Find the volume of the bead with r = 1 and R = 2.
22 2 2
2 2
3 22
3
33
3
4 1 4
3 4
4 1
3 3 3 3 3 3 3 4 33
4 3
x y y
y x y
V y dy
yy
V
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Chapter 7: Techniques of IntegrationSection 7.1: Numerical Integration
Jon Rogawski
Calculus, ETFirst Edition
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The shaded area in Figure 1 cannot be calculated directly using a definite integral, since there is not an explicit antiderivative forInstead, we will rely on numerical approximation using the trapezoidal method
2
2x
e
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
If we divide the interval [a, b] into N even intervals, the area may be found using the Trapezoidal Rule
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
As shown in Figure 3, the area of the trapezoidal segment is equal to the average of the left- and right-RAM areas.
As shown in table one, by increasingthe size of N, we can attain whateverdegree of accuracy we may need.
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 5 illustrates how a mid point estimate rectangle has the samearea as a trapezoid where the top of the trapezoid is tangent to the curve at the midpoint of the interval.
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Example, Page 424Calculate TN and MN for the value of N indicated.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
4
02. 4xdx N
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Example, Page 424Calculate TN and MN for the value of N indicated.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2
18. ln 5xdx N
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Example, Page 424Calculate the approximation to the volume of the solid obtained by rotating the graph about the .
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
823. cos ; 0, ; -axis; 2
y x x M
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
We frequently are concerned with the accuracy of the estimate
obtained using either the trapezoidal or midpoint method. They
may be defined as follows:
Error Errorb bN N N Na aT T f x dx M M f x dx
If we assume f ″ (x) exists and is continuous on our interval, we may use Theorem 1.
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 6 shows how trapezoidal estimates for areas under curvesare more accurate for those with small values of f ″ .
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 6 shows the points we would use in calculating T6 and M6 for an approximation to the area of the shaded region in Figure 8.
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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 10 illustrates how trapezoids provide an underestimate of areas under concave down curves and midpoints provide over-estimates. The opposite holds true for concave up curves.
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Example, Page 424State whether TN or MN overestimates or underestimates the integral and find a bound for the error. Do not calculate for TN or MN.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
210132. ln xdx M
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Example, Page 424Use the Error Bound to find a value of N for which the Error (TN) ≤ 10 – 6.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
5
236. dx
x
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Homework
Homework Assignment #16 Read Section 7.2 Page 424, Exercises: 1 – 11(Odd), 25, 29,
33, 37
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company