Find the volume of the solid obtaining by rotating the region bounded by the curves y=0, Y=sin, 0<=x<=pi; about the line y=-2.
Have you drawn a picture? of course, sin(x)< 2 for all x so the region that is being rotated to form the solid is the region between y= 2 and y= sin(x). For each x, the line segment from y= 2 to y= sin(x), which has length 2- sin(x) rotates to a disk of radius 2- sin(x) and so area $\displaystyle \pi(2- sin(x))^2$. Each "disk" will have thickness dx so the volume is given by
$\displaystyle \pi\int_0^\pi (2- sin(x))^2 dx$.