Calculus derivation 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 with
volume = 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)
= pi(4/3r^3)
which is the familiar
formula: '4/3pi*r^3'.