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When I plug something like this into Mathematica:

$\int_0^{x^2-1} k y \, dy$

I get exactly what I would expect:

$\frac{k}2 (x^2-1)^2 $

However, when I change my bounds ever so slightly, from $x^2-1$ to $1-x^2$ I would expect this:

$\frac{k}2 (1-x^2)^2 $

But I actually end up with the same as before:

$\frac{k}2 (x^2-1)^2 $

I'm at a loss as to what I'm missing. I ran these through WolframAlpha as well and got the same results, so I must be missing some basic rule of integration. For reference, here are the Mathematica commands I'm running:

Integrate[k y, {y, 0, (x^2 - 1)}] Integrate[k y, {y, 0, (1 - x^2)}] 

2 Answers 2

3

Rewrite $(x^2-1)^2$ as $((-1)(1-x^2))^2$ to convince you it's the same.

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    In this case it wasn't so much that I forgot $a-b$ is $b-a$, but that $(x)^2 = (-x)^2$2012-11-08
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The two are the same since $(-x)^2 = x^2$.