Math

Find the volume generated when the area bounded by the curve y^2 = 16x, from x = 0 to x = 4 is rotated around the x-axis.

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  1. using discs:

    v = ∫[0,4] πr^2 dx
    where r=y
    v = ∫[0,4] πy^2 dx
    = π∫[0,4] 16x dx
    = 16π∫[0,4] x dx
    = 16π(1/2 x^2)[0,4]
    = 16π(8-0)
    = 128π

    using shells:

    v = ∫[0,8] 2πrh dy
    where r = y and h = 4-x = 4-y^2/16
    v = 2π∫[0,8] y(4 - y^2/16) dy
    = 2π (2y^2 - y^4/64)[0,8]
    = 2π(128 - 64)
    = 128π

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