of the volume
of a sphere
I once had a math
test for which I needed this formula
. I didn't have it memorized so I tried to derive it. I eventually came up with the correct equation but when I looked at my work later I realized I had a bunch of mistakes
that just happened to cancel each other out. I've reworked the derivation
so that it makes sense. Here it is.
Let A be the region bounded by:
the semi-circle y = |sqrt(r^2 - x^2)|,
the line y = 0,
the line x = -r,
and the line x = r
where r is a constant.
Let V be the volume
of the solid
defined by rotating A about the x-axis. We'll assume (since I haven't figured out how to prove it) that this solid is spherical.
A slice of the solid taken parallel to the y-axis is a cylinder
= pi*y^2*dx. Now, take the sum of the slices from x = -r to r, as dx approaches 0.
pi*y^2 = pi(r^2 - x^2)
The Indefinite Integral
of 'pi(r^2 - x^2)dx' is 'pi(xr^2 - 1/3x^3) + c'
And the Definite Integral
from x = -r to x = r gives us:
V = pi(r^3 - 1/3r^3 + c - (-r^3 + 1/3r^3 + c)) (by the Fundamental Theorem of Calculus
= pi(2/3r^3 + r^3 - 1/3r^3)
which is the familiar formula