Finding area was rigorously defined by Riemann, which is why we call these Riemann Sums. The sums transform into the integral, which is used to find area.
EDIT: I apologize. My Latex coding is not working. Perhaps a mod could point out the error.
[Don't worry about it. LaTeX is pretty complicated. It took me a while to get that working. Hopefully that's what you meant to say ]