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Math Help - Proof of equal area under curve

  1. #1
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    Proof of equal area under curve

    1.) Without evaluating the integrals, prove that the following areas under the function's curves are equal:

    f(x) = \frac{x}{(1+x^4)} on the interval of 0 to 2
    and
    g(x) = \frac{1}{2(1+x^2)}\ on the interval of 0 to 4


    I've no idea where to start on this one... not without evaluating integrals. I do have their graphs to look at, but I'm not seeing anything obvious. Any help is appreciated...
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  2. #2
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    Quote Originally Posted by ebonyscythe View Post
    f(x) = \frac{x}{(1+x^4)} on the interval of 0 to 2
    and
    g(x) = \frac{1}{2(1+x^2)}\ on the interval of 0 to 4
    It's just proving that

    \int_0^2 {\frac{x}<br />
{{\left( {1 + x^4 } \right)}}\,dx} = \frac12\int_0^4 {\frac{{dx}}<br />
{{1 + x^2 }}\,dx} .

    And this does not require big effort, so for the left integral substitute u=x^2 and you're done.
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  3. #3
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    So that's not evaluating the integrals then? That's what I was mostly wondering, but now that I think about it, you're not technically evaluating the integrals, just showing that they are equal to each other...

    Alright, thank you very much. I wanted to make sure there wasn't some "other" way to do it.
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  4. #4
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    Sure, it's not necessary to evaluate them. Just a little substitution shows that they're equal to each other.
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