How to find area under sines without calculus?

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

You can see Roberval’s method in Seventeenth-Century Indivisibles Revisited, by Vincent Julien, pp. 192-194. Here is a link to the section in Google Books.

It’s a complicated geometric argument, and I haven’t worked through the details, but it doesn’t look to me like it’s the same as a Riemann sum. The proof seems to be based on constructing similar triangles with one infinitesimal side, within a circle.

Attribution
Source : Link , Question Author : Lee David Chung Lin , Answer Author : J. M. ain’t a mathematician

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