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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