# How do you find the volume of the bounded region if y = sinx, y = 0 from x = pi/4, x = 3pi/4, revolved around the y-axis?

Jun 19, 2015

Use the shell method.

#### Explanation:

Representative shell has volume: $2 \pi r h \cdot \text{thickness}$

$r = x$, $h = \sin x$ and $\text{thickness} = \mathrm{dx}$

So the volume of the solid is given by:

$V = {\int}_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} 2 \pi x \sin x \mathrm{dx}$

Use integration by parts to get:

V =2pi( sinx-xcosx)]_(pi/4)^((3pi)/4

Then use trigonometry and arithmetic to get:

$V = 2 \pi \cdot \frac{\pi \sqrt{2}}{2} = {\pi}^{2} \sqrt{2}$