# Area under parabola using geometry

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.

#### Solutions Collecting From Web of "Area under parabola using geometry"

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

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.)