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

Question

22229 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-09-10T23:00:31

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

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 $V=4/3pir^3$

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