Riemann Integration in the Euclidean Space

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 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 A enclosed by a parabola and a line.