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

↳Redirected from "Suppose that I don't have a formula for #g(x)# but I know that #g(1) = 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
1 Answer
Konstantinos Michailidis
Sep 10, 2015

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#

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