Volume of a Sphere in Spherical Coordinates

This calculation comes down to an iterated integral:

V =

2pi

0

pi

0

r

0

2 sin ddd

Evaluating the innermost integral yields

V =

2pi

0

pi

0

1

3r3 sin dd.

Evaluating the next integral we get

V =

2pi

0

2

3r3 d.

This gives us the desired formula:

V =4pi

3r3.

1