# Article

# Article

- Mathematics
- Analysis (calculus)
- Integration

# Integration

Article By:

**Graves, Lawrence M. **Formerly, Department of Mathematics, University of Chicago, Chicago, Illinois.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.347900

- Riemann integral
- Antiderivatives
- Elementary methods of integration
- Improper integrals
- Multiple integrals

- Line, surface, and volume integrals
- Functions defined by integrals
- Approximate and mechanical integration
- Other methods of integration

- Integral of Lebesgue
- Other definitions of integration
- Links to Primary Literature
- Additional Readings

**An operation of the infinitesimal calculus which has two aspects.** The roots of one go back to antiquity, for Archimedes and other Greek mathematicians used the “method of exhaustion” to compute areas and volumes. A simple example of this is the approximation to the area of a circle obtained by inscribing a regular polygon of known area, and then repeatedly doubling the number of sides. The areas of the successive polygons are computable with the help of elementary geometry. The limit of the sequence of these areas gives the area of the circle. The area of each polygon can be regarded as being made up of the sum of the areas of triangles with vertices at the center of the circle, and so the process described is a constructive definition of an integral which is the limit of a sum. Modern definitions of integrals as limits of sums are discussed in this article.

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