# Unexpected results when integrating definite integral with variable bounds

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)}]

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Rewrite $(x^2-1)^2$ as $((-1)(1-x^2))^2$ to convince you it's the same.
Folks seem to forget that the negative of $a-b$ is $b-a$. – Lubin Oct 29 '12 at 0:42
In this case it wasn't so much that I forgot $a-b$ is $b-a$, but that $(x)^2 = (-x)^2$ – Finster Nov 8 '12 at 18:01
The two are the same since $(-x)^2 = x^2$.