Volume of Solid of Revolution (Disk/Washer)

Question

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves $x=y−y^2$ and $x=0$ about the y-axis.

Answer 1

The intersection between the curves $x=y-y^2$ and $x=0$ is obtained by considering the equation $$0=y-y^2.$$ Solving the equation, we get $y=0$ or $y=1$. The points of intersection are $(0,0)$ and $(0,1)$. The graph of the required region is given below:

If we rotate the horizontal strip about the $y$-axis, a disk is obtained whose volume is given by $$dV=\pi x^2dy=\pi(y-y^2)^2dy=\pi(y^2-2y^3+y^4)dy.$$ Hence, the required volume of the solid generated is $$V=\int_0^1 dV=\int_0^1\pi(y^2-2y^3+y^4)dy=\pi\bigg[\frac{y^3}{3}-\frac{y^4}{2}+\frac{y^5}{5}\bigg]_{0}^1=\pi\bigg[\frac{1}{3}-\frac{1}{2}+\frac{1}{5}\bigg]=\frac{\pi}{30}.$$