Proof of equal area under curve

**ebonyscythe** · December 3rd 2007, 12:25 PM

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...

**Krizalid** · December 3rd 2007, 12:36 PM

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

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

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

**ebonyscythe** · December 3rd 2007, 12:58 PM

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.

**Krizalid** · December 3rd 2007, 01:00 PM

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

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