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Calculus

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Let R be the square region bounded by y = 2, x = 2, and the
x and y-axes. When R is rotated about the x-axis, what is the volume of the
resulting solid? What is the volume of the solid generated by rotating R about
the y-axis?

  • Calculus -

    first, since it's a square, it does not matter which side is used as the rotational axis.

    geometrically, we have a cylinder of height=2, radius=2

    v = pi * 2^2 * 2 = 8pi

    Analytically,

    v = Integral(pi y^2 dx)[0,2]
    = Integral(4pi dx)[0,2]
    = 4pi x[0,2]
    = 8pi

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