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This is my first post. I´m writing here because I have a doubt regarding a solved problem of double integration present in a book of mine. I don´t speak english yet ( ), but I will try to translate the problem to english. Well, here I go:

"Calculate int[ (x^2 + y^2) dx dy ] over the region D on the first quadrant of the xy-plane limitated by the hyperboles x^2 - y^2 = 1, x^2 - y^2 = 9, x*y = 2 and x*y = 4".

Well, using the following transformation: u = x^2 - y^2, v = x*y; we obtain the following Jacobian: J(u,v) = 1 / ( 2 * (u^2 + 4*v^2)^(1/2) ).

The region of integration Q on the uv-plane is the rectangle: Q = {(u,v) E R² | 1 <= u <= 9 , 2 <= v <= 4}.

So,

int[ (x^2 + y^2) dx dy ] = int[ (u^2 + 4*v^2) * ( 1 / ( 2 * (u^2 + 4*v^2)^(1/2) ) ) du dv ] = (1/2) * int[ du dv ] = 8.

Well, I don´t understand this solution. If you plot the region Q, you will obtain a rectangle lying on the uv-plane, but this rectangle includes (through the previous change of coordinates) the two regions lying on the xy-plane that are limitated by the four hyperboles: the first one lying on the first quadrant, and the second one lying on the third quadrant. But I want only the region on the first quadrant. I can´t see how the definition of the rectangle Q could exclude the region on the third quadrant. And this region must to be excluded, because the change of coordinates must to be injective (otherwise I can´t use the previous method to calculate double integrals).

I don´t know if you could understand what I wrote, or even if my doubt was clearly exposed (in case of my horrible english have been understood). But I thank in advance for any help!

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# A doubt about double integration

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