Method of exhaustion

Definition

The method of exhaustion is an ancient mathematical technique used to find the area under a curve by inscribing and circumscribing polygons whose areas converge to the desired value. It involves taking the limit of the sum of the areas of these polygons as their number increases indefinitely.

5 Must Know Facts For Your Next Test

1. The method of exhaustion was first developed by ancient Greek mathematicians such as Eudoxus and Archimedes.
2. It is considered an early form of integration, where the area under a curve is approximated using geometric shapes.
3. The method relies on creating a sequence of increasingly accurate polygonal approximations for the given region.
4. This technique can be seen as a precursor to modern techniques such as Riemann sums and definite integrals.
5. Archimedes used this method to rigorously prove results like the area of a circle being $\pi r^2$.

Review Questions

Related terms

Riemann Sum:

A method for approximating the total area underneath a curve on a graph, often used as an approach to defining integrals.

Definite Integral:

The evaluation of the integral from one specific point to another, representing the area under a curve between those two points.

Limit:

\text{The value that a function or sequence approaches as input or index approaches some value.}