I presume you have drawn a graph? y= x^2 and y= x+ 2 intersect at (-1, 1) and (2, 4).

Using the shell method, you would rotate a vertical line, at a given "x" around x= 2 so its radius would be 2- x. The shell height will be .x=2

This is on the left side so the radius is x- (-1)= x+ 1.x=-1

Now you are rotating around a horizontal axis (y= 0) so your variable will be y, not x. The radius will be y. As for the length of the "shell", you will need to do it in two parts. from y= 0 up to y= 1, it will be and from y= 1 up to y= 4, it will be [tex]\sqrt{y}- (y- 2)= \sqrt{y}- y+ 2[tex].x axis How?

Again, this is a horizontal axis so your variable will be y. The radius will by 4- y.y=4

I know i have the limits of integration correct but i cant figure out what the correct radius or shell height are. Any guidance is appreciated.