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

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0How do I do it? I think the limits change?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I am not sure if I understand your question right. Here's an example anyway: \[\int\limits_{a}^{b}\int\limits_{c}^{d}f(x,y)dxdy=\int\limits_{c}^{d}\int\limits_{a}^{b}f(x,y)dydx\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0http://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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0But 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Does that make sense?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0If 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Here 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Actually 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So ignore that last image ;p
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