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Changing the order of integration: surefire method?

  1. Apr 19, 2007 #1
    Is there a way to transform the limits of integration for a multivariable integral without appealing to geometrical manipulations? For example:
    [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".
    Last edited: Apr 19, 2007
  2. jcsd
  3. Apr 19, 2007 #2


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    The region of integration is simply the set defined by the system of inequalities

    [tex]a \leq x \leq b,[/tex]
    [tex]y_1(x) \leq y \leq y_2(x),[/tex]
    [tex]z_1(x, y) \leq z \leq z_2(x, y).[/tex]

    So, all you need to do is to formally manipulate these inequalities to put them into the form you want in the new integral.

    I.E. You would first solve this system of inequalities for z, then for y in terms of z, then for x.
  4. Apr 20, 2007 #3
    I would not expect a simple general algorithm since you may be forced to split the integral into several regions: That can happen say when one of the z_i(x,y),y_j(x) functions is not one-to-one.
  5. Apr 20, 2007 #4
    Thanks Hurkyl and gammamcc.
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