# Volume of solid by cross-section question?

1. Feb 3, 2010

### zeion

1. The problem statement, all variables and given/known data

I need to find the region bounded by these curves then find the volume of the solid generated by revolving this region about the x-axis.

y= cscx, x= 1/4pi, x = 3/4pi, y=0

2. Relevant equations

3. The attempt at a solution

So I managed to sketch this region.. but I have trouble finding the anti-derivative at the end.. so it looks like this:

$$V = \pi \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} [(cscx)^2 - (0)^2]dx = \pi \left[ -cotx \right]_{\frac{\pi}{4}}^{\frac{3\pi}{4}} = \pi(-cot(\frac{3\pi}{4})-(-cot(\frac{\pi}{4})) = -3+1 = -2??$$

2. Feb 3, 2010

### Staff: Mentor

Your integral and its antiderivative look fine, but what happened to pi? Your problem doesn't seem to be in the integration, but in evaluating cot(x).
Tip: bring the - outside so that you have -pi(cot(x)), evaluated at 3pi/4 and pi/4.

So you have -pi(cot(3pi/4) - cot(pi/4)).
cot(3pi/4 = -1 and cot(pi/4) = 1.

Now what do you get? It should be positive.