TITLE: Area under parabola using geometry
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: 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.)