How to find area under sines without calculus?

Question

In the section establishing that integrals and derivatives are inverse to each other, James Stewart's Calculus textbook says (pp325--pp326, Sec.4.3, 8Ed): >When the French mathematician Gilles de Roberval first found the area under the sine and cosine curves in 1635, this was a very challenging problem that required a great deal of ingenuity. If we didn’t have the benefit of the Fundamental Theorem of Calculus, we would have to compute a difficult limit of sums using obscure trigonometric identities. It was even more difficult for Roberval because the apparatus of limits had not been invented in 1635. I wonder how Gilles de Roberval did it. Wikipedia and MacTutor do not contain much info on that. How to apply the method of quadrature is exactly the real challenge I suppose. This is mainly a history question, but I'm also curious as to how one would approach this in modern days. Thank you.

Answer 1

If you go to Google and search "gilles de roberval area under the curve" there is an answer in a document "Roberval" from the Arizona State University. It is about the 7th one down.