Let R be the region bounded by y = ln x and the x-axis on the interval [1,2]. Find an integral expression for the volume of the solid obtained by rotating R around
(a) the x-axis
(b) the line x = 2
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 ey + r must equal 2, so the radius of the disk is
r = 2 – ey.
The volume of the disk is
π(2 – ey)2 Δ y.
The variable y ranges from 0 to ln 2 in the region R, so the volume of the entire solid is