Answer

The region R looks like

(a) When we rotate R around the x-axis we get the solid

Slicing perpendicular to the x-axis gives us disks of radius y = ln x.

The volume of the disk at position x is

π (ln x)² Δ x.

Since x goes from 1 to 2 in the region R, the volume of the entire solid is

(b) Rotating R around the line x = 2 produces the solid

Slices perpendicular to the line x = 2 are horizontal disks.

Looking at the disk at height y, we see that e^y + r must equal 2, so the radius of the disk is

r = 2 – e^y.

The volume of the disk is

π(2 – e^y)² Δ y.

The variable y ranges from 0 to ln 2 in the region R, so the volume of the entire solid is