Consider the region bounded by the $y$ axis, $y=3-x^2$, $y=x+x^2$ and $x=-1$. This region is rotated around the $y$ axis to form a solid.
Now I've constructed my shell etc and acquired the integral
$V=2\pi \int_{a}^{b} 3x-x^2-2x^3$
Now if I take the limit from $-1$ to $0$, I get a negative answer whereas if I do it the other way around, I get a positive answer.
Why is this the case? With other shells problems, I've always taken the limits to be from the "smaller number" to the "bigger number" and that's always worked out fine. What is different in this example?
The reason why you're getting a negative answer is that the radius of the shell is $-x$ not $x$, and hence you need to apply a minus sign to the integrand and therefore to the final answer, which is $12\pi$.