Problem is, given the following functions and conditions:
$y = x^9 +x^3 +2$, $y = \cos(x)$, $x = 1$ and $x = 2$
Rotated around $x = 4$.
I know how to do the problem, but the only part I'm really struggling with is the outer and inner radius. How exactly would you know which is which if you didn't know what the graphs looked like?
Let me first repharse your problem: You want to calculate the rotational volume that is bounded by the functions $f(x) = x^9+x^3+2$ and $g(x) = \cos(x)$ in the interval $x=[1,2]$.
As you said the formula is given by
$$ V = \pi \int_1^2 \left|g^2(x)-f^2(x)\right| dx.$$
Here it does not matter at all which one is $g$ and which one is $f$ since squaring the functions makes them both nonnegative and the absolute value is both times the same: $|a-b| = |b-a|$, so you can use $g$ and $f$ interchangeably.