Chapter 8.2, Problem 82E

A family of exponentials The curves y = xe^−ax are shown in the figure for a = 1, 2, and 3.

1. a. Find the area of the region bounded by y = xe^−x and the x-axis on the interval [0, 4].
2. b. Find the area of the region bounded by y = xe^−ax and the x-axis on the interval [0, 4], where a > 0.
3. c. Find the area of the region bounded by y = xe^−ax and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b), where a > 0 and b > 0.
4. d. Use part (c) to show that A(l, ln b) = 4A(2, (ln b)/2).
5. e. Does this pattern continue? Is it true that A(1, ln b) = a²A(a, (ln b)/a)?