Multiple Integral

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

An th-order integral corresponds, in general, to an -dimensional volume (i.e., a content), with  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  that is first performed over a variable  and then performed over a variable  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.

REFERENCES:

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.