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)}]