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I'm wondering about the surface area of the graph of $f(x,y)=xy$ for $x,y$ in an axis-aligned rectangle. The surface area is given by the integral $$ \int_m^n \int_p^q \sqrt{x^2 + y^2 + 1} \, dx \, dy $$ but this doesn't seem to simplify. Can anyone suggest any tricks, or anything in terms of special functions?

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Have you tried using polar coordinates? That is, $(x,y) = (rcos \theta, rsin\theta)?$ After that, you could try a trigonometric substitution on the transformed integral.

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