Yes, because your cross sections are "perpendicular to the x-axis", the thickness, perpendicular to the area, is measured along the x-axis.

Two errors:and the radius of the semi circle is 1/2 the height between y = 8 - x^2 and y = x^2

which gives

then you will have the intergral from x = -2 to x = 2 because this is where they intersect

For some reason I dont think im accurately doing this

You left out the "thickness" which becomes "dx" in the limit. NEVER write an integral without the "dx"!

Also, it is the entire polynomial that is squared. .

Now, what are the limits of integration?