Volumes of Solids of Revolution Method of Cylinders — Question 7

Question 7 – Solution

1. Find the intersections.

Both sides of $\sqrt{x}=x/2$ are nonnegative. Squaring and factoring gives$$x=\frac{x^2}{4}\quad\Longrightarrow\quad x(x-4)=0\quad\Longrightarrow\quad x=0,4.$$Both values satisfy the original equation.

2. Identify radius and height.

Use vertical shells about the $y$-axis. On $[0,4]$, $\sqrt{x}\ge x/2$:$$r(x)=x,\qquad h(x)=\sqrt{x}-\frac x2.$$

3. Set up and expand the shell integral.

$$V=2\pi\int_0^4x\left(\sqrt{x}-\frac x2\right)dx=2\pi\int_0^4\left(x^{3/2}-\frac{x^2}{2}\right)dx.$$

4. Integrate using the power rule.

$$\int x^{3/2}\,dx=\frac25x^{5/2},\qquad\int\frac{x^2}{2}\,dx=\frac{x^3}{6}.$$$$V=2\pi\left[\frac25x^{5/2}-\frac{x^3}{6}\right]_0^4.$$

5. Apply the bounds and simplify.

Since $4^{5/2}=(\sqrt4)^5=32$,$$V=2\pi\left(\frac{64}{5}-\frac{64}{6}\right)=2\pi\left(\frac{192-160}{15}\right)=\boxed{\frac{64\pi}{15}}.$$

All volumes are in cubic units.