Archimedes’ quadrature of the parabola The Greeks solved
several calculus problems almost 2000 years before the discovery
of calculus. One example is Archimedes’ calculation of the area
of the region R bounded by a segment of a parabola, which he did
using the “method of exhaustion.” As shown in the figure, the idea
was to fill R with an infinite sequence of
began with an isosceles triangle inscribed in the parabola, with
area A1, and proceeded in stages, with the number of new triangles
doubling at each stage. He was able to show (the key to the solution)
that at each stage, the area of a new triangle is 1/8 of the
area of a triangle at the previous stage; for example, A2 = 1/8 A1,
and so forth. Show, as Archimedes did, that the area of R is 4/3
times the area of A1.
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