Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

y=e^(-x), y=1, x=2; about y=2.

Don't you mean x=1, x=2; about y=2?

No, actually. It is y=1. I sketched out the solid of the equations, and I believe it's a washer.

To find the volume of the solid obtained by rotating the given region around the line y = 2, you can use the disk method. The disk method involves summing up the volumes of infinitesimally thin disks that make up the solid.

First, let's sketch the region bounded by the curves y = e^(-x), y = 1, and x = 2.

The region looks like a triangle formed by the y-axis, the curve y = e^(-x), and the line y = 1. The point of intersection between y = e^(-x) and y = 1 is (0, 1).

To use the disk method, we need to express the radius of each disk as a function of y. In this case, the radius is the distance between the line y = 2 and the curve y = e^(-x).

Since the axis of rotation is y = 2, the distance from y = 2 to y = e^(-x) is 2 - e^(-x).

Next, we need to determine the limits of integration for y. The solid is bounded by y = 1, so the lower limit of integration is 1. The upper limit of integration is the y-value at x = 2.

To find the y-value at x = 2, substitute x = 2 into the equation y = e^(-x):
y = e^(-2)

Now, we can set up the integral to find the volume:
V = π∫[1, e^(-2)] (2 - e^(-x))^2 dy

Evaluate this definite integral to find the volume of the solid.