Single Variable Calculus, Chapter 6, 6.3, Section 6.3, Problem 32

$\displaystyle \int^{\frac{\pi}{4}}_0 2 \pi (\pi - x)(\cos x - \sin x) dx$ represents a volume of a solid. Describe the solid.

We can see from the equation that shell method was used with vertical strips. The distance of these strips from a line that it will be revolved in is $\pi - x$. If you revolve this length about such line, you'll get a circumference of $C = 2 \pi (\pi - x)$. By these data, we assume that the line we are talking is $x = \pi$. Also, the height of these strips resembles the height of the cylinder by $\displaystyle H = y_{\text{upper}} - y_{\text{lower}} = \cos x - \sin x$. Thus,


$\begin{equation} \begin{aligned} V =& \int^{\frac{\pi}{4}}_0 C(x) H(x) dx \\ \\ V =& \int^{\frac{\pi}{4}}_0 2 \pi (\pi - x) (\cos x - \sin x) dx \end{aligned} \end{equation}$


In short, the expression is obtained by rotating the region bounded by $\displaystyle y = \cos x$ and $\displaystyle y = \sin x \, 0 \leq x \leq \frac{\pi}{4}$ about the line $x = \pi$.