Question

How to find the volume of a sphere using integration?

Answer 1

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`