Asked by ani
                Let S be a region bounded by the curve y=x+cosx  
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.
            
        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.
Answers
                    Answered by
            Steve
            
    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
    
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
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