The region is roughly trapezoid shaped, with vertices at (0,1)(0,2)(2,1)(5,2)

With shells, we have to integrate on x, since the shell thickness is dx.

From x=0-2, we just have a rectangle (which, revolved is just a cylinder of radius 2, height 1), and for x=2-5, we have shells of height 2-y.

Since x=1+y^2, y = √(x-1)

v = 2π*2*1 + ∫[2,5] 2πrh dx

where r = x and h = 2-√(x-1)

v = 4π + 2π∫[2,5] x(2-√(x-1)) dx

= 4π + 2π(59/15)

= 178/15 π

just to check, we can use discs, and we have

v = ∫[1,2] πr^2 dy

where r = x

v = π∫[1,2] (y^2+1)^2 dy

= 178/15 π