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I was working out this example in Jon Rogawski multivariable calculus book:

enter image description here

My question is: why didn't he take all possible regions that satisfy the conditions: $$1\le xy\le4,1\le y/x\le4$$

which include the region in the third quarter ??

Also, if we choose only the region in the first quadrant to be integrated, then after transformation of coordinate, there are two areas satisfy transformation:

$$[1,2]\times [1,2]$$ as the example, and also: $$[-1,-2]\times [-1,-2]$$

So why don't we include it in integration ?

Mohamed Mostafa
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    To avoid such confusion, the author should have specified that $\mathcal D$ resides in the first quadrant. My guess is he simply forgot to do so, or assumes $x>0$ and $y>0$ elsewhere in the text, perhaps when he first introduced the transformation $G$. – user170231 Aug 08 '16 at 00:09

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