# Thread: Proof of equal area under curve

1. ## 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...

2. Originally Posted by ebonyscythe
$f(x) = \frac{x}{(1+x^4)}$ 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

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

And this does not require big effort, so for the left integral substitute $u=x^2$ and you're done.

3. 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.

4. Sure, it's not necessary to evaluate them. Just a little substitution shows that they're equal to each other.