Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$\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 $0 < v < e^2$

share|cite|improve this question
please add the homework tag, and show what you tried, if you tried nothing, try something and come back again :) – Dominic Michaelis Mar 8 '13 at 8:37
Please check if I've edited it correctly. – Ishan Banerjee Mar 8 '13 at 8:38
I did do some work but I was unsure about it. Will post it here. Thanks :) – Kj Tada Mar 8 '13 at 8:40
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
up vote 0 down vote accepted

Here is a plot of the region $S$:

enter image description here

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?

share|cite|improve this answer
Thanks for providing me with the graph this helps a lot. However, I am not sure what you mean when you ask "Where are the limits in x for each piece?" Do you mean the limit of y=e^2x, y=4/x, y=1/x, y=x.I will update what I have done so far but Math typing is new to me and its taking sometime for me to type Math. – Kj Tada Mar 8 '13 at 9:30
I am asking you where the various curves intersect each other. By looking at the plot, you can tell which intersections you need to solve. – Ron Gordon Mar 8 '13 at 9:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.