How to find the volume of a sphere using integration?

1 Answer
Sep 18, 2014

Since a sphere with radius r can be obtained by rotating the region bounded by the semicircle y=sqrt{r^2-x^2} and the x-axis about the x-axis, the volume V of the solid can be found by Disk Method.

V=pi int_{-r}^r(sqrt{r^2-x^2})^2dx

by the symmetry about the y-axis,

=2piint_0^r(r^2-x^2)dx

=2pi[r^2x-x^3/3]_0^r

=2pi(r^3-r^3/3)

=4/3pir^3