Question

Evaluate the integral SS( e^((x+y)/(x-y)) )dA where the region is the trapezoidal region with vertices (1,0), (2,0), (0,-2), and (0,-1)

Answer 1

First, draw a diagram of the area of integration

Answer 2

Then figure out how you'd like to integrate it. One way would be to break it up into four triangles.

Answer 3

Do you have any idea how you would use "transformation of variables" to solve this kind of problem?

Answer 4

Yes, the integrand gives you some hints here of what to try.

How about
u = x + y
v = x - y

Figure out what the region of integration looks like in u,v-coordinates

Answer 5

Ok so the region is once again a trapezoid. Solving for x and y in terms of u and v ends up being x = .5(u+v) and y = .5(u-v)

Answer 6

Oh, I forgot to include the jacobian, which it explains later in the chapter. Sorry to bother.

Answer 7

Let me know when you're back and we'll talk through this. My initial suggestion wasn't the optimal choice.