Posted by **matt** on Saturday, January 31, 2009 at 4:53pm.

Use integrals to prove that the volume of a sphere of radius

R is equal to (4/3)(pi)R^3

- calculus -
**Damon**, Saturday, January 31, 2009 at 5:06pm
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

- calculus -
**Damon**, Saturday, January 31, 2009 at 5:17pm
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]

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