Calculus/Volume of solids of revolution

Revolution solids

A solid is said to be of revolution when it is formed by rotating a given curve against an axis. For example, rotating a circle positioned at (0,0) against the y-axis would create a revolution solid, namely, a sphere.

Calculating the volume

Calculating the volume of this kind of solid is very similar to calculating any volume: we calculate the basal area, and then we integrate through the height of the volume.

Say we want to calculate the volume of the shape formed rotating over the x-axis the area contained between the curves $f(x)$ and $g(x)$ in the range [a,b]. First calculate the basal area:

$|\pi f(x{)}^2-\pi g(x{)}^2|$

And then integrate in the range [a,b]:

$\underset{a}{\overset{b}{\int }}|\pi f(x{)}^2-\pi g(x{)}^2|dx=\pi \underset{a}{\overset{b}{\int }}|f(x{)}^2-g(x{)}^2|dx$

Alternatively, if we want to rotate in the y-axis instead, $f$ and $g$ must be invertible in the range [a,b], and, following the same logic as before:

$\pi \underset{a}{\overset{b}{\int }}|{f^{-1}(x)}^2-{g^{-1}(x)}^2|dx$