Let U be the region in the first quadrant that is bounded above by the curve y=e−x, below by the curve y=e−2x, and on the left by the y–axis. The region extends infinitely far to the right.
Answer the following questions.
(a) Write, but do not evaluate, an integral expression that represents the exact area of U. Clearly indicate any limits of integration that involve infinity.
(b) Evaluate the integral from part (a) to find the exact area of U.
(c) Determine the exact x–coordinate of the centroid (center of mass) of U assuming uniform density. Write any necessary integrals and show all work.