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Find the volume of the solid of revolution by roating the region R about the y-axis. Write the answer as a multiple of pi.
y= sqrt(x), y=x^2, in the first Quadrant

  • Calculus -

    The curves intersect at (0,0) and (1,1).

    Using shells,

    v = ∫2πrh dx [0,1]
    where r = x and h=√x-x^2
    v = 2π∫x(√x-x^2) dx [0,1]
    = 3π/10

    Using discs,

    v = ∫π(R^2-r^2) dy [0,1]
    where R=√y and r=y^2
    v = π∫(y-y^4) dy [0,1]
    = 3π/10

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