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Math Help - Dividing an integrand by a strictly positive function

  1. #1
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    Dividing an integrand by a strictly positive function

    Hi all,

    I've got a really nasty integral that I want to show is zero. It's of the form (x,y are functions of time)

    \oint_C{\frac{f(x,y) dx + g(x,y) dy}{\dot{x}^{2} + \dot{y}^{2}}

    I'm thinking I need to use Green's Theorem and the algebra for this would be massively simplified if I could discard the denominator, which is positive everywhere. So my question is, is the following true? How would I show it, and why not if it isn't?

    \int{f dx} = 0 \Rightarrow \int{\frac{f}{g^2} dx} = 0
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  2. #2
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    Re: Dividing an integrand by a strictly positive function

    Hy karatekid.

    Consider f(x) = sin(x) and g(x) = x from 2*pi to 4*pi. The Integral of f(x) = 0 over that region but now the integral of f(x)/g^2(x) is not zero:

    http://www.wolframalpha.com/input/?i=int+sinx%2Fx^2+dx+from+2*pi+to+4*pi

    Basically you can choose any function that scales the origin in such a way the symmetry of the parts that were balanced (i.e. add to zero) have become un-balanced with the right kind of scaling.
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