Calc

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=3x^2, x=1, y=0, about the x-axis

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  1. using discs of thickness dx,
    v = ∫[0,1] πr^2 dx
    where r=y=3x^2
    v = ∫[0,1] π(3x^2)^2 dx = 9π/5

    using shells of thickness dy,
    v = ∫[0,3] 2πrh dy
    where r=y and h=1-x = 1-√(y/3)
    v = ∫[0,3] 2πy(1-√(y/3)) dy = 9π/5

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  2. Vol = π ∫ y^2 dx from 0 to 1
    = π ∫ 9x^4 dx from 0 to 1
    = π[ (9/5)x^5 ] from 0 to 1
    = π [(9/5)(1) - 0]
    = 9/5π

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