Question

Use integrals to prove that the volume of a sphere of radius
R is equal to (4/3)(pi)R^3

Answer 1

well the really easy way is to assume that you know the area of a sphere is 4 pi r^2
integral dr (4 pi r^2)
= 4 pi r^3 / 3 = (4/3) pi r^3

Answer 2

Otherwise radius of plane parallel to x,y axis as a function of height above the center plane is
r = R cos theta
area of circular plane at height z= R sin theta = pi r^2 = pi R^2 cos^2 theta
so hemisphere above axis
integral dz pi R^2 cos^2 theta
but dz = d theta R cos theta
so
integral d theta R cos theta pi R^2 cos^2 theta
from theta = 0 to theta = pi/2
pi R^3 d theta cos^3 theta
integral of cos^3 = sin - (1/3) sin^3
so
pi R^3 [sin theta - (1/3 sin^3 theta]
at pi/2 - at 0
pi R^3 [ 2/3]
that is for the upper hemisphere, multiply by 2 to include the bottom half
pi R^3 [4/3]