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I am talking about double integrals. I have seen many examples of how to reverse the integration order when current order is too difficult to calculate. But I still don't fully understand how to do it. I would like you to help me understanding this.

Suppose the integral:

∫∫f(x,y) dydx

and a region defined by:

x = 0 to 1

y = 1 - x^2

I want to change the order of integration from dydx to dxdy. I have some understanding of this, but I don't know if it is correct. I will ask some questions in order to understand.

1 - In the more external integral the limits have numbers, so we can get a number out of the integration. Reversing the order we will first integrate in x and then in y (y goes in external integral). To figure out what are the new y limits (in the external integral) I just need to plug x = 0 and x = 1 in (1 - x^2) equation to find the down limit and the upper limit, respectively, is that right? So we will integrate from y = 1 to y = 0?

2 - We know that y depends on x accordingly with y = 1 - x^2. But we need to figure out how does x depends on y, and then we plug this relation in the internal integral, is that right?

3 - The most important question. How do find how does x depends on y? Is there a rule to that? Could I just find the inverse function? I know that we can plot the region in the xy plane and try to find how x depends on y looking at the graph and figuring out, but isn't there a rule for that? A method for doing this? A method that will always works (or at least in the major cases).

Thank you for your help!

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# Understanding how to reverse the integration order

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