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Mathematics LibreTexts

3: Multiple Integrals

The multiple integral is a generalization of the definite integral with one variable \[\int f(x) dx\] to functions of more than one real variable, for example, \[\iint f(x, y) \,dx\,dy\] for an indefinite double integral or \[\iiint f(x, y, z)\, dx\,dy\,dz\] for an indefinite  triplet integral. For definite multiple integrals, each variable can have different limits of integration.

Thumbnail: Double integral as volume under a surface \(z = 10 − x^2 − y^2/8\). The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated. Image used with permission (Public Domain; Oleg Alexandrov)