## Volume of Cone Using Volumes of Revolution

The formula for the volume of a cone can be derived using volumes of revolution. For a cone of base radius
$r$
and height
$h$
we can rotate the line
$y= \frac{r}{h} x, \; 0 \le x \le h$
$x$
\begin{aligned} V &= \pi \int^h_0 y^2 dx \\ &= \pi \int^h_0 (\frac{r}{h} x)^2 dx \\ &= \pi \frac{r^2}{h^2} \int^h_0 x^2 dx \\ &= \pi \frac{r^2}{h^2} [\frac{x^3}{3} ]^h_0 \\ &=\frac{1}{3} \pi r^2 h \end{aligned}