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Calculus

Find the volume of the solid obtained by rotating the region bounded by y= x^2, y=0, and x=3 about the x-axis.

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  1. V = π∫(y^2) dx from a to b

    = π∫x^4 dx from 0 to 3
    = π[x^5/5] from 0 to 3
    = π(243/5 - 0) = 243π/5 cubic units

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  2. or, using shells, you can do

    V = ∫[0,9] 2πrh dy
    where r = y and h = 3-x
    = 2π∫[0,9] y(3-√y) dy
    = 2π(3/2 y^2 - 2/5 y^(5/2)) [0,9]
    = 2π(3/2 * 81 - 2/5 * 243)
    = 2π(243/10)
    = 243π/5

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