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Please bear with me because I have only little experience in using codes to construct the symbols for the equations. The question is:

Determine the average value of $f(x,y) = x^2 y^2$, in the region $$R: a\le x\le b, c\le y\le d,$$ where $a+b=5, ab=13, c+d=4, cd=7.$

The formula for average value is

$$\dfrac{\displaystyle\iint f(x,y)dA}{(b-a)(d-c)}.$$

Please use only elementary calculus, and no complex analysis. Much appreciated!

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    Two points: 1. Is this a homework question? If yes, please mark it as such. 2. Solving the constraints and $a$ and $b$ gives complex results. Are you sure these conditions are correct as given?2012-03-27
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    oh sorry, that should be double integral2012-03-27
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    I'm an instructor in a tutorial center, we just found this problem in a sample exam of a 1st year college student in a calculus subject. Yes I'm aware that it gives complex result, this is why I'm asking if there is some sort of manipulation in the solving the problem and not directly evaluating the integral.2012-03-27
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    I don't think so - the expression $a \le x \le b$ does not make sense with $a,b$ complex.2012-03-27
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    As stated the problem simply makes no sense; it must be defective.2012-03-27
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    You have a point sir. But I think there is no typo error with the question because the values of c and d are also imaginary. My point is that, I think the student's instructor has a solution for the question, the problem is he did not showed it. Anyway, thanks for the thought though.2012-03-27
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    Even if you use the fact that all of $a,b,c,d$ are imaginary, the question still does not make any sense. On the other hand, if you evaluate the integrals as line integrals over $\mathbb C$, you do get a (negative) real solution since the real parts cancel out, and the product of the imaginary parts is real.2012-03-27
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    Even if it doesn't make sense I've computed formally an average value of 12.2012-03-27
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    Note that you can get a double integral with correct spacing using `\iint`.2012-03-27
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    This is also a pretty idiotic way to present a region of integration, even if the solution to the system of equations had $a,b,c,d \in \mathbb{R}$.2012-03-27

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