  The so-called Riemann sums have their origin in the efforts of Greek
mathematicians to find the center of gravity or the volume of a solid body.
These researches led to the method of exhaustion, discovered by Archimedes and
described using modern ideas by MacLaurin in his \textit{Treatise of Fluxions}
in 1742. At this times the sums were only a practical method for computing an
area under a curve, and the existence of this area was considered geometrically
obvious. The method of exhaustion consists in almost covering the space
enclosed by the curve with $n$ geometric objects with well-known areas such as
rectangles or triangles, and finding the limit (though this topic was very
blurry at these early times) when $n$ increases. One of its most remarkable
application is squaring the area $\mathcal{A}$ enclosed by a parabola and a
line.
