the region r is the area enclosed by the functions y = 2x and y = x² shown below. find the volume of the…

Explanation:

Step1: Find intersection points

Set $2x = x^{2}$, then $x^{2}-2x=0$, factoring gives $x(x - 2)=0$. So $x = 0$ and $x = 2$. When $x = 0,y = 0$; when $x = 2,y=4$.

Step2: Use the method of cylindrical - shells (or the disk/washer method in terms of $y$). When rotating about the $y$-axis, using the disk/washer method in terms of $y$. Solve $y = 2x$ for $x=\frac{y}{2}$ and $y = x^{2}$ for $x=\sqrt{y}$.

The volume formula for rotating about the $y$-axis using the disk/washer method is $V=\pi\int_{a}^{b}(R^{2}-r^{2})dy$. Here, $R=\sqrt{y}$, $r = \frac{y}{2}$, and $a = 0$, $b = 4$. So the volume is $V=\pi\int_{0}^{4}(\sqrt{y}-\frac{y}{2})^{2}dy$.

Answer:

$\pi\int_{0}^{4}(\sqrt{y}-\frac{y}{2})^{2}dy$