EDIT: Sorry, not sure how to delete this, but I figured out my problem. I should be taking the limits with respect to t, not x.

"Find the area enclosed by the x-axis and the curve x=1+e^t, y=t-t^2."

I think my problem is setting up the integral. The integral I set up yields the wrong answer. Here's my reasoning:

First of all the limits of the integral should be the corresponding values of x for which y = 0. These will be 2 and 1+e. Now y=t-t^2 and dx=e^t, so the integral of ydx = integral of (e^t)(t-t^2)dt, with limits 2 and 1+e.

Sorry if that's unclear, I really need to learn basic LaTeX (I think it's called) to express math better but I'm in a hurry right now.