We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without using calculus, with directly using geometry. I think it would be very interesting challenge, but I am not able to think of a way out.
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Using a similar transformation used by Archimedes for sphere and cylinder, show equivalence of slice of the curve at point $x$ to the area of a pyramid slice. The total area will be equal to the volume of the pyramid of unit base and height. This is however equivalent to calculus (under disguise.)
It depends a little bit what you mean by geometry. If you can see "geometrically" that stretching the function horizontally by a factor of $2$ should double the value of the integral, and if you can see "geometrically" that the integral of a sum of two functions should be the sum of the integrals, then there is such a proof, and I spell it out here:
https://mathoverflow.net/questions/114738/integrating-powers-without-much-calculus/114843#114843