Volumes of Solids of Revolution Method of Cylinders — Question 9

Question 9 – Solution

1. Identify shell radius and height.

Use vertical slices about the $y$-axis on $[0,1]$:$$r(x)=x,\qquad h(x)=\arctan x.$$Angles are measured in radians.

2. Set up the volume integral.

$$V=2\pi\int_0^1 x\arctan x\,dx.$$

3. Apply integration by parts.

Let $u=\arctan x$, $dv=x\,dx$, so $du=dx/(1+x^2)$ and $v=x^2/2$:$$\int x\arctan x\,dx=\frac{x^2}{2}\arctan x-\frac12\int\frac{x^2}{1+x^2}\,dx.$$

4. Simplify the remaining rational function.

$$\frac{x^2}{1+x^2}=1-\frac1{1+x^2}.$$Since $\int(1+x^2)^{-1}dx=\arctan x$, an antiderivative is$$F(x)=\frac{x^2}{2}\arctan x-\frac x2+\frac12\arctan x.$$

5. Apply the bounds and simplify.

Using $\arctan1=\pi/4$ and $\arctan0=0$,$$F(1)-F(0)=\frac\pi8-\frac12+\frac\pi8=\frac\pi4-\frac12.$$$$V=2\pi\left(\frac\pi4-\frac12\right)=\boxed{\pi\left(\frac\pi2-1\right)}.$$

All volumes are in cubic units.