Volume of Solid of Revolution/Parametric Form

Theorem
Let $x: \R \to \R$ and $y: \R \to \R$ be real functions defined on the interval $\closedint a b$.

Let $y$ be integrable on the (closed) interval $\closedint a b$.

Let $x$ be differentiable on the (open) interval $\openint a b$.

Let the points be defined:
 * $A = \tuple {\map x a, \map y a}$
 * $B = \tuple {\map x b, \map y b}$
 * $C = \tuple {\map x b, 0}$
 * $D = \tuple {\map x a, 0}$

Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by:
 * $\set {\tuple {\map x t, \map y t}: a \le t \le b}$

Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$).

Then the volume $V$ of $S$ is given by:
 * $\ds V = \pi \int_a^b \paren {\map y t}^2 \map {x'} t \rd t$