Volume of Solid of Revolution

Theorem
Let $f: \R \to \R$ be a real function which is integrable on the interval $\closedint a b$.

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

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

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:
 * $\ds V = \pi \int_a^b \paren {\map f x}^2 \rd x$

Proof

 * VolumeOfSolidOfRevolution.png

Consider a rectangle bounded by the lines:
 * $y = 0$
 * $x = \xi$
 * $x = \xi + \delta x$
 * $y = \map f x$

Consider the cylinder generated by revolving it about the $x$-axis.

By Volume of Cylinder, the volume of this cylinder is:
 * $V_\xi = \pi \paren {\map f x}^2 \delta x$