First, graph the region.
The curves intersect when 8-x2 = x2. So, they intersect when x = 2.
The region lies in quadrant 1. It is bounded on the left by the vertical line x = 1, on the right by the vertical line x = 2, above by the curve
y = 8-x2, and below by y = x2.
Take a typical vertical slice of the region of width Δx at a distance x from the y-axis.
Rotate the slice about the y-axis. The resulting shell has volume 2π(radius)(height)(thickness).
radius = x
height = (8-x2)-x2 = 8-2x2
thickness = Δx
Volume of shell = 2πx(8-2x2)Δx = 4π(4x-x3)Δx
Volume of solid = 4π∫(from x=1 to x=2)(4x-x3)dx