Math 208
Cylindrical and spherical coordinates problems
Set up and evaluate each of the following in either cylindrical or spherical coordinates,
whichever is more appropriate:
1. , where Q is the region with , inside the sphere , and
Q
x dV
∫
0
x
≥
2 2 2
16
x y z
+ + =
below the cone .
2 2
z x y
= +
2.
2 2 2
2 2
2 2
2
2 2 2
02
2
x x x y x
x y
x y
x x
dz dy dx
− +
+
− +
− −
∫ ∫ ∫
3.
2 2 2
2 2 2
2 8 2 8 1
0 0
y y z
yx y z
dx dz dy
− − −
+ +
∫ ∫ ∫
4.
2
22 2
0 4 1
2 2
y z y
y y x y
dx dz dy
− −
− − +
∫ ∫ ∫
5. Find the mass and z-coordinate of the center of mass of the object inside the sphere
and below the cone , if the density is
2 2 2
4
x y z z
+ + =
2 2
3 3
z x y
= +
.
2 2 2
1
( , , )
x y z
x y z
δ
+ +
=
