Find the volume when $x=\csc y, \;y=\pi/4\,$ to $\,y= 3\pi/4$

Question

Find the volume when $x=\csc y,\,$ from $\,y=pi/4,\,$ to $\,y= 3\pi/4$, about the y-axis.

I tried to integrate $V = \displaystyle\int_{\pi/4}^{3\pi/4} \pi\csc^2(y)\,dy$.

But the answer was with negative.

Why ?

The answer in the book is $V = 2\pi$

Answer 1

Your set-up of the integral is correct. I suspect you may have omitted a sign, or otherwise incorrectly integrated $\csc^2 y$. Recall that $$\dfrac {d}{dx}\left(\cot y\right) = -\csc^2 y$$ and so we have that $$\int \csc^2 y\, dy = -\cot y + C$$

This means that

$$\begin{align} V = \pi \int_{\pi/4}^{3\pi/4} \csc^2 y \,dy & = -\pi \cot y \;\Big|_{\pi/4}^{3\pi/4} \\ \\ & = -\pi\Big(\cot(3\pi/4) -\cot (\pi/4)\Big) \\ \\ & = -\pi( -1 - 1) \\ \\ &= 2\pi\end{align}$$