Post on 05-Jul-2018
8/16/2019 TUTORIAL_Triple Integral.pdf
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TUTORIAL – TRIPLE INTEGRALS
1. Evaluate the following:
a) 3 2 1
2
0 1 0
xy zdxdy dz∫ ∫ ∫
Ans: 21/4
b) 1 2
0 0
2 x y
x xyzdz dy dx∫ ∫ ∫
Ans: 5/8
c) dxdzdyze3
0
1
0
z1
0
y2
∫ ∫ ∫ −
Ans:1/3(e3-1)
TRIPLE INTEGRAL IN CYLINDRICAL COORDINATES
1. Evaluate the iterated integrals
a) ∫ ∫ ∫ −
3
0
2
0
r 4
0
2
ddr dzzr
π
θ θ Ans:9
2π b) ∫ ∫ ∫
4
0
2
0
4
r dr ddzr
π
θ Ans: π 3/64
2) Evaluate ( )2
2 2 2
2 4 2
2 2
2 4
x
x x y
x y dzdydx
−
− − − +
+∫ ∫ ∫ . Ans: 5/16π
3) Use cylindrical coordinates to evaluate the following triple integrals
( )∫ ∫ ∫−−
−− ++
3
3
x9
x9
3
yx
222
2 22dxdydzyx Ans: π 10/243
4) Find the volume of the region enclosed by the cylinder 2 2 4 x y+ = and the planes 0 z =
and 4 y z+ = . Ans: π 16
5) Using the cylindrica l coordinates system, find the volume of the solid bounded below by
the sphere 2 2 2 2 x y z+ + = and above by paraboloid 2 2 z x y= + . Ans: π 30/13
6) Find the volume of the region enclosed by the paraboloid 2 2 z x y= + and the plane 9= z .
Ans: 2/81π
7) Find the volume of the solid bounded by the paraboloid 2 24 z x y= − − and the xy-plane.
Ans: π 8
8) Find the volume of the solid E that lies within the cylinder 2 2 1 x y+ = , below the plane
4 z = and above the paraboloid 2 21 z x y= − − . Ans: 2/7π
9) Use triple integral to find the volume of the given solid enclosed by the paraboloid22 yxz += and the plane z=16. Ans: π 128
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10) Find the volume of the solid bounded above by 2 4 x y z+ + = in the first oc tant.
Ans: 5.33
11) Find the volume of the solid bounded above by 6 3 2 z x y= − − in the first oc tant.
Ans:6
TRIPLE INTEGRAL IN SPHERICAL COORDINATES
1. Evaluate the iterated integrals
a) ∫ ∫ ∫602
0
3
0
2 dddsinπ π
φ θ ρ φ ρ
Ans :
− 2
3
12
9π
b) ∫ ∫ ∫π π
π θ φ ρ φ ρ 2
02
2
1
2 dddsin
Ans : π 3/14
2) Evaluate
2 22
2
42 4
2 2 2 2
2 04
x y x
x
z x y z dzdydx
− −−
− − −
+ +∫ ∫ ∫ . Ans: 3/32π
3) Using the spherical coordinate, find the volume of portion of the sphere2 2 2
9 x y z+ + = lying in the first oc tant. Ans: 2/9π
4) Using the spherical coordinate, find the volume of portion of the sphere
2 2 216 x y z+ + = and below by the cone 22 yxz += . Ans: ( )22
3
64−
π