Volume of Solid of Revolution

Theorem
Let $f: \R \to \R$ be a real function which is integrable on the interval $\left[{a \,.\,.\, b}\right]$.

Let the points be defined:
 * $A = \left({a, f \left({a}\right)}\right)$
 * $B = \left({b, f \left({b}\right)}\right)$
 * $C = \left({b, 0}\right)$
 * $D = \left({a, 0}\right)$

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

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:
 * $\displaystyle V = \pi \int_a^b \left({f \left({x}\right)}\right)^2 \, \mathrm d x$

Proof

 * VolumeOfSolidOfRevolution.png

Consider a rectangle bounded by the lines:
 * $y = 0$
 * $x = \xi$
 * $x = \xi + \delta x$
 * $y = f \left({x}\right)$

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

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