# How do you find the volume of a cone using an integral?

A cone with base radius $r$ and height $h$ can be obtained by rotating the region under the line $y = \frac{r}{h} x$ about the x-axis from $x = 0$ to $x = h$.
$V = \pi {\int}_{0}^{h} {\left(\frac{r}{h} x\right)}^{2} \mathrm{dx} = \frac{\pi {r}^{2}}{{h}^{2}} {\int}_{0}^{h} {x}^{2} \mathrm{dx}$
$= \frac{\pi {r}^{2}}{h} ^ 2 {\left[{x}^{3} / 3\right]}_{0}^{h} = \frac{\pi {r}^{2}}{{h}^{2}} \cdot {h}^{3} / 3 = \frac{1}{3} \pi {r}^{2} h$