Not sure exactly how to get started on this:
Find volume of the solid by rotating the region bounded by $y=x^3$, $x=0$, $y=0$ and $y=-8$ around $y=3$.
I know that the graph should be a cubic function, and the shaded portions should be from $y=0$ to $y=-8$ inside the lower portion of the cubic function $y=x^3$, but the rotation around $y=3$ is what is throwing me off. Up until this point it has just been "rotate around y-axis" or "rotate around x-axis".
Does this mean that the height of the cylinder should be $3-x^3$ or $3-y$? If so, this is what I came up with so far, in terms of y (most likely completely wrong, but hey):
$y=x^3 \Rightarrow 3=x^3$ ; $x=3^\frac{1}{3} \Rightarrow x=y^\frac{1}{3}$
So circumference is: $2\pi y^\frac{1}{3}$
Height: $3-y$
Volume: $V=\int_{-8}^0 \left(2\pi y^\frac{1}{3}\right)\left(3-y\right)$
So far so good?