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Fubinis Theorem

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The integral of a continuous function over an elementary region

$\displaystyle V:\ a_j(x_1,\ldots,x_{j-1})\le x_j \le b_j(x_1,\ldots,x_{j-1}) $

can be computed by the iterated integral

$\displaystyle \int\limits_V f\,dV = \int_{a_1}^{b_1} \int_{a_2(x_1)}^{b_2(x_1)}... ..._{n-1})}^{b_n(x_1,\ldots,x_{n-1})} f(x_1,\ldots,x_n)\,dx_n\cdots dx_2 dx_1 \,. $

If the integration limits are constants then the iterated integral is independent of the ordering of the variables provided the integration limits are ordered correspondingly, e.g. for double integrals

$\displaystyle \int_{a}^b \int_{c}^d f(x,y) \ dy dx = \int_{c}^d \int_{a}^b f(x,y) \ dx dy \ .$

If the integration limits are not constant it may be possible that the elementary region cannot be written as an elementary region for a different ordering of the variables. Then one has to split the region. In each case one has to adapt the integration limits to the different ordering of the variables.

see also:

[Examples]
  automatically generated 5/30/2011