MATHEMATICS 201-BNK-05
Advanced Calculus
Martin Huard
Winter 2015
XIX - Triple Integrals in Cylindrical
and Spherical Coordinates
1. Sketch the solid whose volume is given by the integral and evaluate the integral.
a)
c)
3
2
3
0
0
r
  

 
0
2sin 
0

rdzd dr
4 r
1 r2
2
b)
rdzdrd
d)


4
2
  

2
0


  
0
4
0
4
0
 2 sin  d d d
2cos
sec
 2 sin  d  d d
2. Use cylindrical coordinates to find the volume of the solid S.
a) S is enclosed by the paraboloid z  x 2  y 2 and the plane z  16 .
b) S lies inside the cylinder x 2  y 2  1 and the sphere x2  y 2  z 2  9 .
c) S is bounded by cone x2  y 2  z 2 and the paraboloid z  5  4 x2  4 y 2 .
d) S is the solid between the cylinders x 2  y 2  x and x 2  y 2  2 x , and bounded below
by z  0 and above by y  z  3 .
3. Use spherical coordinates to find the volume of the solid S
a) S is enclosed by the sphere x2  y 2  z 2  16 and the cone z  3x 2  3 y 2 .
b) S is the wedge cut from a sphere of radius a by two planes that intersect along a
diameter at an angle of 4 .
c) S is inside the hemisphere z  4  x 2  y 2 and outside the sphere x2  y 2  z 2  z .
4. Evaluate the following integrals using either Cartesian, cylindrical or spherical coordinates,
whichever seems more appropriate.
a)
9 x2
3
 
3
 9 x
2

9
x2  y 2
xdzdydx

3
2
 
1 x 2
c)

x 2  y 2 dV where S is the region that lies inside the cylinder x2  y 2  16 and
1
1 0

1 x 2  y 2

b)
0
e
 x2  y 2  z 2
dzdydx
S
between the planes z  4 and z  5 .
d)   x 2  y 2  z 2  dV where S is the region in the first octant bounded by the sphere
S
x2  y 2  z 2  25 , the cone z  x 2  y 2 and the cone z  2 x 2  y 2 .
Math BNK
e)
XIX - Triple Integrals in Cylindrical and Spherical Coordinates
 xdV
where S is enclosed by the planes z  0 , x  y  z  3 and by the cylinders
S
x 2  y 2  4 and x 2  y 2  9 .
f)
 sin
x 2  y 2  z 2 dV where S is the solid within the sphere x2  y 2  z 2  49 , above
S
the xy-plane and below the cone z  x 2  y 2 .
5. Find the centroid of S.
a) S is the solid inside the sphere x2  y 2  z 2  2 and the cone z  x 2  y 2 .
b) S is the solid bounded by the paraboloid z  4 x 2  4 y 2 and the plane z  a , a  0 .
c) S is bounded by the xy-plane and the hemispheres
y  9  x2  z 2
and
y  16  x 2  z 2 .
6. Find the center of gravity for the solid S.
a) S is enclosed by the sphere
x2  y 2  z 2  4z
and
the
density
is
  x, y, z   x 2  y 2  z 2 .
b) S is the solid that is bounded by the cylinder x 2  y 2  1, the cone z  x 2  y 2 , and
the xy-plane if the density is   x, y, z   z .
c) S is bounded above by the plane z  y and below by the paraboloid z  x 2  y 2 if the
density is   x, y, z   x 2  y 2 .
7.
Find the moments of inertia for the hollow cylinder a12  x 2  y 2  a22 , 0  z  h , assuming a
constant density   x, y, z    .
8. Find the moments of inertia for the solid lying between two concentric hemispheres of radii r
and R, where r  R , assuming a constant density   x, y, z    .
9. Evaluate the improper integrals, if they converge.
a)
 e
 x2  y 2  z 2
dV
3
b)
 1  x
2
S
c)
Winter 2015
2
4 x2
0
 4 x
 
2
1
dV
 y2  z2


0
where S is the inside of the cone z  x 2  y 2
1
dzdydx
1 x  y2  z2
2
Martin Huard
2
Math BNK
XIX - Triple Integrals in Cylindrical and Spherical Coordinates
ANSWERS
1. a) 9
b)
2. a) 128
3. a)
128
3
 32 32
b) 36  64 32
 64 33
4. a) 0
e)
64
3
65
4
b)
 a3
b)

f)
3 2 1
5. a)  0, 0, 8 


16
6. a)  0, 0, 7 
6
3
 3e
c) 134  329
c)
7
3
c)
31
6
d)
d)
c) 384

3
9
4
d) 500 5  625 2
2 14sin  7   47 cos  7   2 
b)
 0,0, 23a 
c)
525
 0, 525
296 , 296 
b)
 0,0, 815 
c)
 0, 74 , 4924 
7. I x  4h  a24  a14   3h  a22  a12  , I y  4h  a24  a14   3h  a22  a12  , I z  2h  a24  a14 
3
3
8. I x  815  R5  r 5  , I y  815  R5  r 5  , I z  815  R5  r 5 
9. a) 8
Winter 2015
b) Diverges
c)
2
2

Martin Huard

5 1
3