x= -y + 3 (you may have a typo in your posting)
-y+3 = y^2 + 1
y^2 + y - 2 = 0
(y+2)(y-1) = 0
The points of intersection are (5, -2) and (2, 1).
Draw a picture to decide the method, the area consists of regions in the first quadrant and in the fourth.
While you could use a combination of disk/washer methods, I would recommend shell:
x_left = y^2 + 1
x_right = -y + 3
Height = x_right - x_left =
circumference = 2 pi distance = 2 pi y as y ranges from -2 to 1.
So Volume = Integral (2pi y Height) dy with interval of y values -2 to 1.