Answer:
f(x) = π ((3 + sin x)² − 1²)
Explanation:
This solid is hollow. So if we cut a cross section, the result will be a "washer", or a disk with a hole in it. The volume of the washer is the volume of the disk minus the volume of the hole:
dV = π R² h − π r² h
dV = π (R² − r²) h
In this case, the outside radius, or radius of the disk, is:
R = y₂ − (-1)
R = 2 + sin x − (-1)
R = 3 + sin x
The inside radius, or radius of the hole, is:
r = y₁ − (-1)
r = 0 − (-1)
r = 1
And of course, the thickness is h = dx.
Therefore:
dV = π ((3 + sin x)² − 1²) dx
So f(x) = π ((3 + sin x)² − 1²).