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Evaluate the (double) integral of $\frac{xy^2}{(4x^2 + y^2)^2}$ over the finite region enclosed by $y= x^2$ and $y = 2x$. My question is: I have tried this by the method of iterated integrals but then I noticed that at $(0,0)$ the function is undefined.

How would you go about solving this? Also is the region here unbounded?

Many thanks

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    You edited the question, but now it still doesn't mention any integral?2012-10-14

2 Answers 2

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As @joriki noted in his second post, we have a bounded region (see below) :

enter image description here

Now try to intersect the functions $x^2=y=2x$ to find out that $0\le x\le 2$ and so you get, as plot tells us, that: $\int_0^{x=2}\int_{x^2}^{y=2x}\frac{xy^2}{(4x^2+y^2)^2}dydx$ It is not hard exploiting the substitution $y=a\tan(\theta)$ for $\int\frac{y^2}{(a^2+y^2)^2}dy$ where $a=2x$.

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    @amWhy: Thanks my friend. Thanks. :-)2013-11-27
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On the first question: The integrand grows like $1/r$ at the origin, but the width of your region also decreases as $r$, so you should be OK. I'd integrate over $x$ first, since the numerator contains the inner derivative of the denominator.

On the second question: The problem explicitly says to integrate over the finite region enclosed by the curves. "Bounded" is just the more formal term for what they call "finite", so no, the region is not unbounded.

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    @CAF : I put the @ symbol here to make sure you read my comment above.2012-10-17