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$\begingroup$

Here is a proof of integration by parts: http://www.math.ufl.edu/~mathguy/year/S10/int_by_parts.pdf

But I don't understand how it works. Specifically, I don't understand why $\int_r^s u \, dv$ equals one of the areas (and likewise for $\int_p^q v \, du$).

To me it looks like this only works if $g$ is the inverse of $f$...

Thanks!

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    I share your opinion. To me this picture looks more like a proof of Young's inequality(http://en.wikipedia.org/wiki/Young's_inequality)2012-12-09

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