# Find the volume of the solid?

#### kensington

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.

#### undefined

MHF Hall of Honor
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.
I always draw a sketch when solving these. How far did you get?

#### HallsofIvy

MHF Helper
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
$$\displaystyle \pi\int_0^\pi (2- sin(x))^2 dx$$.