Volumes of Solids of Revolution Method of Cylinders — Question 10

Question 10 – Solution

1. Identify shell dimensions and set up the integral.

Use vertical shells about the $y$-axis, with $0\le x\le1$:$$r(x)=x,\qquad h(x)=\ln(1+x^2).$$$$V=2\pi\int_0^1x\ln(1+x^2)\,dx.$$

2. Substitute and change the bounds.

Let $u=1+x^2$. Then $du=2x\,dx$, so $x\,dx=du/2$. The bounds change as follows:$$x=0\Rightarrow u=1,\qquad x=1\Rightarrow u=2.$$

3. Rewrite the volume in terms of $u$.

$$V=2\pi\cdot\frac12\int_1^2\ln u\,du=\pi\int_1^2\ln u\,du.$$

4. Integrate the logarithm by parts.

Choose $a=\ln u$ and $db=du$, giving $da=du/u$ and $b=u$:$$\int\ln u\,du=u\ln u-\int u\frac1u\,du=u\ln u-u.$$

5. Evaluate the endpoints.

$$\begin{align*} V&=\pi\left[u\ln u-u\right]_1^2\\&=\pi\bigl[(2\ln2-2)-(\ln1-1)\bigr]\\&=\pi(2\ln2-1)=\boxed{2\pi\left(\ln2-\frac12\right)}. \end{align*}$$

All volumes are in cubic units.