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Triangle integral ∫∫dxdyf(x*y) how to reduce to one dimension?

  1. Feb 14, 2012 #1
    I meet with a triangle integral where x+y≤1, and function is dependant only on x*y. I am wondering if there any possibility to relate ∫∫dxdyf(x*y)=∫d(x*y)f(x*y)g(x*y) or something similar? Or maybe there are some assumptions needed to relate like this?
  2. jcsd
  3. Feb 14, 2012 #2


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    I'm not sure this is what you want. But in the integral $$
    \int_0^1\int_0^{1-x^2}f(xy)\, dydx$$you could change the inner variable from ##y## to ##w## with the substitution ##w=yx,\ dw = x dy## giving$$
    \int_0^1\frac 1 x \int_0^{x-x^2}f(w)\, dwdx$$
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