Written homework problem 13
Assigned 10/23 and due 10/27
(From the spring 2011 take-home exam 3.)
This problem is about evaluating the integral
Z B −x
e − e−2x
I = lim
dx.
B→∞ 0
x
[ This is an improper integral, and I want you to pay careful attention to this. There is no nice
antiderivative for the function above. ]
(a) Find a function g(y) so that
−x
e
−e
−2x
Z
=
2x
g(y) dy.
x
This means that I =
R ∞ R 2x
0
x
g(y)
x
dy dx .
(b) Switch the order of integration in
Z
0
B
Z
x
2x
g(y)
dy dx.
x
You should get two pieces. Explain your reasoning with words and equations and pictures.
(c) Evaluate one of the pieces explicitly from (b) explicitly, and show that the remaining piece
tends to zero as B → ∞. Use this to determine the value of the integral I.
1