CPM Homework Help : CALC3RD Problem 8-62

GEOMETRY PROOF: VOLUME OF A SPHERE

Another way to view a sphere is as a semicircle rotated about an axis. Use a generic semicircle such as $y = \sqrt { r ^ { 2 } - x ^ { 2 } }$ to prove that the volume of a sphere with radius $r$ is $\frac{4}{3}\pi r^3$ .

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.