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SOLVED:Find the volume of the solid generated by revolving about the y axis the area between the first arch of the cycloid x=θ-sinθ, y=1-cosθand the x axis. Use the shell method. From Fig. 41-14, V =2 π∫θ=0^θ-2 π x y d x=2 π∫0^2 π(θ-sinθ)(1-cosθ)(1-cosθ) d θ =2 π∫0^2 π(θ-2 θcosθ θcos^2 θ-sinθ 2 sinθcosθ-cos^2 θsinθ) d θ =2 π[(3)/(4) θ^2-2(θsinθ cosθ) (1)/(2)((1)/(2) θsin2 θ (1)/(4) cos2 θ) cosθ sin^2 θ (1)/(3) cos^3 θ]0^2 π =6 π^3 cubic units Fig. 41-14 (Figure Cant Copy)
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