I am given two curves $y=2-x^2$ and $y=x^2$, from which the bounded region is to be rotated about $x=1$. I have drawn out the shape and see that the two parabolas intersect each other at $(-1,1)$ and $ (1,1) $. However, it seems like the equation for the radii will change at $y=1$, the points of intersection. Using the cylindrical shells method I would be integrating with respect to $x$, so I am not sure how I factor this change into my integral.
Drawing the region we are rotating is very useful. Draw also the vertical line $x=1$.
Because of what you pointed out, computing volume by Slicing is a nuisance, and Cylindrical Shells is (are?) the way to go.
Draw a thin vertical strip "at" $x$, and rotate the ppart of the strip between the two curves about the line $x=1$. The radius of the cylindrical shell is $1-x$. The height of the shell at $x$ is $(2-x^2)-x^2$. Now one can write down the appropriate integral, and compute.