Question

Find the volume formed by rotating the region enclosed by:
y=2sqrt(x) and y=x about the line y=4

Answer 1

The black length represents the outer radius, and the blue one represents the inner radius.
Using the disk method, the volume is
\[\begin{align*}V&=\pi\int_0^4\left((4-x)^2-(4-2\sqrt x)^2\right)\;dx\\
&=\pi\int_0^4\left((16-8x+x^2)-(16-8\sqrt x+4x)\right)\;dx\\
&=\pi\int_0^4\left(x^2-12x+8\sqrt x\right)\;dx\\
\vdots \end{align*}\]