Multiple Integral

A multiple integral is a set of integrals taken over n>1 variables, e.g.,


An nth-order integral corresponds, in general, to an n-dimensional volume (i.e., a content), with n=2 corresponding to an area. In an indefinite multiple integral, the order in which the integrals are carried out can be varied at will; for definite multiple integrals, care must be taken to correctly transform the limits if the order is changed.

In traditional mathematical notation, a multiple integral of a function f(x,y) that is first performed over a variable y and then performed over a variable x is written


In the Wolfram Language, this would be entered as Integrate[f[x, y], {x, x1, x2}, {y, y1[x], y2[x]}], where the order of the integration variables is specified in the order that the integral signs appear on the left, which is opposite to the actual order of integration.

See also

Definite Integral, Double Integral, Fubini Theorem, Indefinite Integral, Integral, Monte Carlo Integration, Multivariable Calculus, Repeated Integral, Triple Integral

Explore with Wolfram|Alpha


Berntsen, J.; Espelid, T. O.; and Genz, A. "An Adaptive Algorithm for the Approximate Calculation of Multiple Integrals." ACM Trans. Math. Soft. 17, 437-451, 1991.Kaplan, W. "Double Integrals" and "Triple Integrals and Multiple Integrals in General." §4.3-4.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 228-235, 1991.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Multidimensional Integrals." §4.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 155-158, 1992.

Referenced on Wolfram|Alpha

Multiple Integral

Cite this as:

Weisstein, Eric W. "Multiple Integral." From MathWorld--A Wolfram Web Resource.

Subject classifications