Multiple Integral
المؤلف:
Kaplan, W
المصدر:
Double Integrals" and "Triple Integrals and Multiple Integrals in General." §4.3-4.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley
الجزء والصفحة:
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23-8-2018
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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], ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/MultipleIntegral/Inline8.gif" style="height:14px; width:5px" />x, x1, x2![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/MultipleIntegral/Inline9.gif" style="height:14px; width:5px" />, ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/MultipleIntegral/Inline10.gif" style="height:14px; width:5px" />y, y1[x], y2[x]![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/MultipleIntegral/Inline11.gif" style="height:14px; width:5px" />], 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.
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