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March 24, 2017

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Revolve the region bounded by y=x and y=x^2 about the y axis. In cubic units, the resulting volume is?

  • Calculus - ,

    You can use discs, integrating along y:

    V = Int(pi (R^2 - r^2) dy)[0,1]
    where R = y and r = sqrt(y)
    = pi*Int(y - y^2)dy[0,1]
    = pi(1/2 y^2 - 1/3 y^3)[0,1]
    = pi(1/2 - 1/3)
    = pi/6

    Or, you can use shells, integrating along x:

    V = Int(2pi*r*h dx)[0,1]
    where r = x h = x-x^2
    = 2pi*Int(x(x-x^2) dx)[0,1]
    = 2pi(x^2 - x^3 dx)[0,1]
    = 2pi(1/3 x^3 - 1/4 x^4)[0,1]
    = 2pi(1/3 - 1/4)
    = 2pi(1/12)
    = pi/6

  • Calculus - ,

    Revolve the region bounded by y = 4x and y = x2 about the y-axis. In cubic units, the resulting volume is

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