Question

Doubled Iterated Integrals: Change the order of integration?

Answer 1

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

Answer 2

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

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

Does that make sense?

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

So ignore that last image ;p