# Surface area of a quadratic surface patch

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?

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.