There are many kinds of integrals, with the most famous being the Riemann integral which is taught in elementary calculus classes.
The motivation behind the Riemann integral is to find the area between curves by using rectangles of infinitely small width. When i first studied calculus, i was fascinated by this formulation and thought it was indeed ingenious.
My question is: what are the limitations of the riemann integral? What could be the link or the motivation to be able to "create" new ones like lebesgue, or darboux?