Revolution about the y axis with shells

Problem

Compute the volume obtained by revolving $y=\sqrt{x}$ on $[0,4]$ about the $y$ axis, using the cylindrical shell method.

Solution

With shells, $V = 2\pi\displaystyle\int_a^b x\,f(x)\,dx$ with $f(x)=\sqrt x$: $V = 2\pi\int_0^4 x\sqrt{x}\,dx = 2\pi\int_0^4 x^{3/2}\,dx = 2\pi\left[\frac{2}{5}x^{5/2}\right]_0^4 = 2\pi\cdot\frac{2}{5}\cdot 32 = \boxed{\dfrac{128\pi}{5}}.$

Links

Topics: Integral
Concepts: Shell method · Volume of revolution
Skills: Compute · Integrate
Exercise types: Integral computation