CPM Homework Help : CALC Problem 8-61

GEOMETRY PROOF: VOLUME OF A SPHERE

Another way to view a sphere is as a semi-circle rotated about an axis. Use a generic semi-circle such as to prove that the volume of a sphere with radius $r$ is $\frac { 4 } { 3 }π r^3$ .

Hint:

Set up and evaluate an integral showing what happens when you rotate the arbitrary semicircle about the $x$ -axis.

Note:
$r =$ radius of the sphere, which is a constant.
$y =$ rotating radii, which are not constant.