Calculus/Volume of solids of revolution

From Wikibooks, the open-content textbooks collection

Current revision (unreviewed)

← Volume	Calculus	Arc length →
	Volume of solids of revolution

[edit] 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.

[edit] 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:

| π f (x) 2 - π g (x) 2 |

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

$\int_a^b |\pi f(x)^2 - \pi g(x)^2|\,dx = \pi \int_a^b |f(x)^2-g(x)^2|\,dx$

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:

$\pi \int_a^b |{f^{-1}(x)}^2-{g^{-1}(x)}^2|\,dx$