
13.7 Spherical Coordinates
Spherical
 ≥ 0,  is the distance of the point to the origin.
 is the positive angle between the positive x axis and
the line segment

OP.
0 , 
is the positive angle between the
positive z axis and the line segment OP.

Conversion equations:

z   cos
r   sin 
x 2  y 2  z 2  2
x   sin  cos 
y   sin  sin 
x 2  y 2   2 sin 2 
Ex. Graph   3.

3
Ex. Graph   4 .

3


Ex. Graph
4 .

Ex. Graph    .

Ex. Convert the equation 6x = x2 + y2 to spherical
coordinates.
Ex. x2 + y2 + z2 = 1
Ex. z = √2
Ex. (x – 1)2 + y2 + z2 = 1
Ex. Convert   3 to Cartesian coordinates.


Ex.   3



f (,, ) dV   f ( ,, )  2 sin  ddd
E
Find
E

and
 in the yz plane
 measures down from the positive z-axis.
 measures out from the origin through the region.

 is measured on the xy plane.
Ex. Change to spherical coordinates
3
 
9x 2
3  9x 2

0
9x 2 y 2
z x 2  y 2  z 2 dzdydx
Ex. Set up integrals to find the volume of the solid
bounded by x2 + y2 = z2 and x2 + y2 = 4.
Ex. Evaluate
 y
E
dV where E is a hollow sphere whose
outside radius is √2 and the inside radius is 1.

Ex. Set up integrals to find the volume of the solid bounded
above by x2 + y2 + z2 = 4z and below by x2 + y2 = z2.
Ex. Set up integrals to find the z coordinate of the center of
mass of the solid in the previous example.
Ex. Set up integrals to find the volume of the solid inside
x2 + y2 + z2 = 2 and outside x2 + y2 = 1.
Ex. Set up integrals to find the volume of the solid bounded
above by z = x2 + y2 and z = 3.
Do: 1. Find the volume of the sphere with radius a.
2. Set up an integral to find the volume of just the cone in
the example using x2 + y2 + z2 = 4z and below by x2 + y2 = z2.
3. Set up integrals to find the volume under the cone and
within the sphere in the same example.
2.
3.