# The method of exhaustion

### Archimedes finds the area of a parabolic segment by exhausting it with triangles, anticipating key ideas of the calculus by two millennia. We shall explore exhaustion and some paradoxical failures.

### Archimedes and quadrature of the parabola

Archimedes (3rd century BC) set himself the task of calculating the area of a parabolic segment, the region between a parabola and a linear chord crossing it, as shown in the figure here.

His treatise, the *Quadrature of the parabola*, develops the geometric theory of parabolas and contains two proofs of his soluti…

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