How do you do this question?

f(x) = π ((3 + sin x)² − 1²)

Step-by-step 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).