Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 1 + sec(x), −π 3 ≤ x ≤ π 3 , y = 3; about y = 1

Using the washer method, the volume is given by the integral

where 3 - 1 = 2 is the distance from y = 3 to the axis of revolution, and similarly (1 + sec(x)) - 1 = sec(x) is the distance from y = 1 + sec(x) to the axis. The integrand is symmetric about x = 0, so the integral "folds" in on itself, and the integral from -π/3 to π/3 is twice the integral from 0 to π/3.

So the volume is