Question

Find the volume of the solid generated by revolving the region bounded by y equals 5 StartRoot Bold sin x EndRoot​, y equals 0 commaand x 1 equals StartFraction pi Over 3 EndFraction and x 2 equals StartFraction 3 pi Over 4 EndFraction about the​ x-axis.

Answer 1

Looks like the region in question is the one bounded by
`y=5√(\sin x)` and the
`x`-axis, between
`x=\frac\pi3` and
`x=\frac{3\pi}4`. Revolving about the
`x`-axis is trivial using the disk method:

`\displaystyle\pi\int_(\pi/3)^(3\pi/4)(5√(\sin x))^2\,\mathrm dx=25\pi\int_(\pi/3)^(3\pi/4)\sin x\,\mathrm dx=\boxed{\frac{25(1+\sqrt2)\pi}2}`