## anonymous 5 years ago Doubled Iterated Integrals: Change the order of integration?

1. anonymous

How do I do it? I think the limits change?

2. anonymous

I am not sure if I understand your question right. Here's an example anyway: $\int\limits_{a}^{b}\int\limits_{c}^{d}f(x,y)dxdy=\int\limits_{c}^{d}\int\limits_{a}^{b}f(x,y)dydx$

3. anonymous

http://www.math.umn.edu/~nykamp/m2374/readings/doubleintchange/ Very bottom, the limits, a,b,c, and d which are 0,1,x, 1, respectively become 0,1,0,y respectively, making the integral possible. I don't understand how that happened.

4. anonymous

But if one of the limit pairs are in terms of the second variable, you have to rewrite the limits in terms of the other variable.

5. anonymous

So originally you were integrating x from 0 to 1, and y from the line y=x to the line y=1. To rewrite it in terms of x then y, you would be integrating y from 0 to 1, and x from the line x=0 to the line x = y.

6. anonymous

Does that make sense?

7. anonymous

If you had tried instead to integrate y from 0 to 1, and x from y to 1 you would not have been integrating over the same domain.

8. anonymous

Here are some illustrations which may help. The green lines are the lines y=1 and y=x, the yellow lines show the area we are integrating the function over (and the direction). The third image shows what we'd be integrating over if we just naively swapped out x with y in our limits.

9. anonymous

Actually that's a terrible illustration because x=1 is the intercept of the line, so if you integrate over that you'd have a domain with no area.

10. anonymous

So ignore that last image ;p