Calculus

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.

x = 7y^2, y ≥ 0, x = 7; about y = 2

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  1. The region has vertices at
    (0,0) (7,0) (7,1)

    So, using shells of thickness dy, we have

    v = ∫[0,1] 2πrh dy
    where r = 2-y and h = 7-x = 7-7y^2
    v = ∫[0,1] 2π(2-y)(7-7y^2) dy = 91π/6

    Or, using discs (washers) of thickness dx,

    v = ∫[0,7] π(R^2-r^2) dx
    where R=2 and r=2-y=2-√(x/7)
    v = ∫[0,7] 2π(2^2-(2-√(x/7))^2) dx = 91π/6

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