TOPICS
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
![int_(x_1)^(x_2)[int_(y_1(x))^(y_2(x))f(x,y)dy]dx=int_(x_1)^(x_2)int_(y_1(x))^(y_2(x))f(x,y)dydx.](https://mathworld.wolfram.com/images/equations/MultipleIntegral/NumberedEquation2.svg) |
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 IntegralExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Multiple IntegralCite this as:
Weisstein, Eric W. "Multiple Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultipleIntegral.html