Volumes of revolution about 2 curves

Question

I'm fairly okay with volumes of revolutions in general but what if I have to rotate it about 2 curves?

The question here asks:

"The region bounded by the given curves is rotated about a specific axis. Set up, but do not evaluate, an integral which gives the volume of the resulting solid by any method."

$y=4x-x^2$,$y=8x-2x^2$ about $x=-2$ and then about $y=-5$

If this asked me how to rotate about the line $x=-2$, that would be fine. How about if I am asked to rotate the curve about $x=-2$ and then $y=-5$? How would I go about doing that?

Answer 1

The wording is confusing, but I'm pretty sure it's simply two separate questions. In other words, they are not asking you to create a solid as the result of rotating over two axes one after another. Think of this as two separate questions:

1. rotate the given region about the axis $x=-2$;

2. rotate the given region about the axis $y=-5$.

Answer 2

With these problems, I always find drawing a graph of both functions to be helpful. Rotating about a $y=$ line, you use the washer method with the given functions. Rotating about a $x=$ line, you will need to change the equations to functions of $y$ rather than $x$.