Volume of revolution about the x-axis

Statement

The region between the curve $y=\sqrt{x}$, the $x$-axis and the lines $x=0$ and $x=4$ is rotated one full turn about the $x$-axis. Compute the volume of the resulting solid.

Solution

Disc method: $V=\pi\int_0^4 \big(\sqrt{x}\big)^2\,dx=\pi\int_0^4 x\,dx=\pi\Big[\frac{x^2}{2}\Big]_0^4=\pi\cdot 8=8\pi.$ Numerically $V\approx 25{.}13$.

$\boxed{V=8\pi\approx 25{.}13}$