Find volume of solid by rotating about x-axis

Question

R is the region bounded by the curves y=tan(x), y=cot(x), and the x-axis on the interval [0, pi/2]. Find the volume of the solid obtained by revolving R around x-axis.

I have tried everything, I am having trouble setting this problem up.

Answer 1

First, start by graphing to get an idea of your region.

From $0$ to $\frac{\pi}{4}$ our region is bounded by $y=\tan(x)$ and the $x$ axis. So the volume of just rotating that region will be:

$$\pi\int_{0}^{\frac{\pi}{4}} \tan^{2}(x)\space\text{d}x$$

The rest of our region from $\frac{\pi}{4}$ to $\frac{\pi}{2}$ is bounded by $y=\cot(x)$ so the volume of rotating just that region will be.

$$\pi\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^{2}(x)\space\text{d}x$$

Find each integral than sum them up.