Let S be a region bounded by the curve y=x+cosx

Question

and the line y=x as shown in the given figure.
Find the volume of the solid generated when S is rotated about the x-axis.
Find the volume of the solid generated when S is rotated about the y-axis.

Answer 1

I assume you want just the region in QI, since otherwise the axis of rotation is inside part of the region.

So, we want the area whose vertices are (0,0), (0,1) and (π/2,π/2)

Around the x-axis, we have, using discs,

v = ∫[0,π/2] π ((x+cosx)^2 - x^2) dx

shells are not feasible, since we cannot solve for x in terms of y.

Around the y-axis, we need shells, so we can integrate along dx:

v = ∫[0,π/2] 2πx((x+cosx)-x) dx