Question about Shell method 
I drew the graph and its 3 dimensional shape. I'm confused on what the Radius would be in this problem. What should I look for when finding this shape's radius?
 A: You have drawn the shape, and the line $x=4$. That gets us close to the end. Take a point $x$ between $-2$ and $-1$. Draw the vertical line at $x$, up to the curve $y=x^{-4}$. Now take a tiny positive "number" $dx$, and draw a vertical line at $x+dx$, up to the curve.
The thin strip between these vertical lines is rotated about the line $x=4$. About how far is the strip from $x=4$? That's your radius.
The distance is $4$ plus the distance from $x$ to $0$. Since $x$ is negative, this distance is $|x|$, or, better, $-x$. 
Thus the radius is $4-x$. The height of the cylindrical shell is $x^{-4}$. This makes the volume of the shell roughly $2\pi (4-x)(x^{-4})\,dx$. It follows that our volume is
$$\int_{-2}^{-1}2\pi(4-x)(x^{-4})\,dx.$$
For the integration, note that $(4-x)(x^{-4})=4x^{-4}-x^{-3}$. 
Remark: Because of a medical condition (allergy to negative numbers) I would almost automatically reflect across the $y$-axis, and rotate the region from $x=1$ to $x=2$ about the line $x=-4$. The integral is then
$$\int_1^2 2\pi(x+4)x^{-4}\,dx.$$
This is quite a bit more pleasant. 
