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

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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#