# Change of Variables to compute integrals

$\int \int e^{xy} dx dy$ over a region $S$ where $S$ is in the first quadrant. $x>0, y>0$ of the $xy$ plane bounded by the curves $xy=1, xy=4$ and the line $y=x, y=e^{2}x$

This is a HW question I have been given for a advanced Calculus class but I am not sure how to go about solving it.

Update: Iam planning to set $u =xy$ so that $1 < u < 4$ but I am not sure what range to choose for v should it be $0 < v < e^2$

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The best thing you can do is draw an accurate picture of the region $S$. – Ron Gordon Mar 8 '13 at 8:45
If you are doing the same problem I'm doing I think you should clarify that the one boundary is not y=e^(2x) but rather y=(e^2)x. Think about how to put these coordinates into another plane, maybe a u,v plane where the boundaries make a nice rectangle instead of some odd shape like it is now. For example if the limits were something like u=a to u=b and v=c to v=d where u and v were functions of x and y, and a,b,c,d were constants, you would have a really nice shape to integrate over. – user65752 Mar 8 '13 at 9:54
thanks, I have made the change.I was having a hard time bcoz I had the question wrong myself. :) I will update what my final soln is toverify it withyou. – Kj Tada Mar 8 '13 at 10:11
Iam planning to set u =xy so 1<u<4 but I am not sure what range to choose for v should it be 0<v<e^2 – Kj Tada Mar 8 '13 at 10:23
I'm assuming the ending of that was e^2 - x ? That's close, you have the right idea. When you arrange for your function v the range will be obvious. So you have u(x,y) = xy, good. v(x,y) = ? so that you have constants as your boundaries? Having those constants is important. Trying v(x,y) = y-x is really messy because of that e^2, but you do need to isolate a constant on one side of the equation, and x and y on the other as v(x,y). I'll give you a hint, the limits for v will be from 1 to e^2 . – user65752 Mar 8 '13 at 10:34

Here is a plot of the region $S$:
If you integrate over $x$, there are 3 pieces to this region. The first is between $y=e^{2 x}$ and $y=1/x$. The second is between $y=4/x$ and $y=1/x$. The third is between $y=4/x$ and $y=x$. Where are the limits in $x$ for each piece?