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Finding the Product of Integrals

  1. Jul 8, 2012 #1
    Is there a formula for calculating the product of integrals, something like:

    [itex]\left(\int_a^b f(x) dx\right) \times \left(\int_c^d g(y) dy\right)[/itex]

    when there is no closed-form expression for F(x) and G(y).

    Actually, the functions are almost identical,

    [itex] f(x) = x^p e^{-x} \text{ and } g(y) = y^q e^{-y}[/itex]

    where p, q are algebraic expressions.

    [itex] F(x) = -\Gamma(x, p) \text{ and } G(x) = -\Gamma(y, q)[/itex]

    and [itex] \Gamma(x, p) [/itex] is defined as another definite integral with an almost identical integrand.

    Thus, is there a way to multiply definite integrals (without knowing the antiderivative) to form one (double?) integral?

    Thanks
     
  2. jcsd
  3. Jul 8, 2012 #2
    If the limits are equal, yes. Then it is converted as follows:
    [tex]\int^{a}_{b}f(x)dx \cdot \int^{a}_{b}g(y)dy = \int^{a}_{b}\int^{a}_{b}f(x)g(y)\,dy\,dx[/tex]
     
  4. Jul 8, 2012 #3
    Is the new integral a double integral then?
     
  5. Jul 8, 2012 #4
    Yes it is.
     
  6. Jul 10, 2012 #5

    haruspex

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    There's no requirement that the limits match. As long as the variables and ranges involved are completely independent, you can always combine them into a double integral (in either order). Just make sure you keep track of which independent variable goes with which range.
     
  7. Jul 10, 2012 #6

    HallsofIvy

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    That's "Fubini's theorem"
     
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