Draw a picture! Your region of integration is a triangle. The inner integral you should think of as going from one function to another, and the outer integral you should think of as going from one number to another. What do you get?
Evaluate the following integral:
I'm thinking the only way I can get through this one is if I switch the order in which we integrate. To do this, I'm going to need to fix the bounds on the integrals.
and I'm not sure what I should do about the bottom bound of the inner integral, i.e.
The bounds on the outter integral should stay the same I think.
Can somebody clarify my confusions before I continue?
I still don't see how I'm suppose to integrate e^(x-x^2).
Oops. I see I didn't make myself clear. Your idea of switching the order of integration is essential to solving this problem. I'm merely trying to help you see what the new limits should be. Before you switch, you're integrating first from the function x = 0 to x = (1-y)/2, and then from the number y = 0 to y = 1.
After you switch, you're integrating first from the function y = ? to y = ?, and then from the number x = ? to x = ?
Fill in the blanks using your triangle.
That should be the whole problem.
Thanks, I didn't think of actually drawing the region and thinking.
Funnily enough, I end up confusing myself if I draw the function - it used to infuriate my lecturers hahaha. I manipulate inequalities instead.
At the moment you have and .
From the first inequality:
So that means one of your new regions of integration is .
You also know
From the second inequality, we know , so therefore your second region of integration is
Therefore, reversing the order of integration gives: