Is there a way to transform the limits of integration for a multivariable integral without appealing to geometrical manipulations? For example:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_a^b \int_{y_1(x)}^{y_2(x)} \int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z) \; dz \; dy\; dx \rightarrow \int_c^d \int_{y_3(z)}^{y_4(z)} \int_{x_3(y,z)}^{x_4(y,z)} f(x,y,z) \; dx \; dy \; dz[/tex]

But how does one determine the limits for the second integral (most importantly, the functions [tex]x_3(y,z)[/tex] & [tex]x_4(y,z)[/tex], etc.) without actually resorting to an illustration of the function and its integration subset? Clearly illustrations are of no use in higher dimensions. Are there any algorithms that generate these interval transformations?

P.S: For simplicity, let us assume that [tex]f(x,y,z)[/tex] is continuous, bounded, and generally "nice".

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# Changing the order of integration: surefire method?

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