Question

How do I use polar coordinates to find the volume of a sphere of radius `r`?

Answer 1

The volume is `V=4/3pir^3`

Explanation:

The equation of a sphere is `x^2+y^2+z^2=r^2`

From the equation we get

`z=+-sqrt(r^2-(x^2+y^2)`

The volume of the sphere is given by

`V=2intint_(x^2+y^2<=r)sqrt(r^2-x^2-y^2)dA`

Using polar coordinates `x=rcosa, y=rsina` and substituing

to the integral above

`V=2int_0^(2*pi)int_0^rsqrt(r^2-a^2)rdrda`

Which is calculated easily giving `V=4/3pir^3`