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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.

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  automatically generated 5/30/2011