use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. sketch the region and a typical shell.

x=1+y^2, x=0, y=1, y=2

3 answers

  1. The region is roughly trapezoid shaped, with vertices at (0,1)(0,2)(2,1)(5,2)

    With shells, we have to integrate on x, since the shell thickness is dx.

    From x=0-2, we just have a rectangle (which, revolved is just a cylinder of radius 2, height 1), and for x=2-5, we have shells of height 2-y.

    Since x=1+y^2, y = √(x-1)

    v = 2π*2*1 + ∫[2,5] 2πrh dx
    where r = x and h = 2-√(x-1)
    v = 4π + 2π∫[2,5] x(2-√(x-1)) dx
    = 4π + 2π(59/15)
    = 178/15 π

    just to check, we can use discs, and we have

    v = ∫[1,2] πr^2 dy
    where r = x
    v = π∫[1,2] (y^2+1)^2 dy
    = 178/15 π

  2. just one more question how did you get the y=squareroot of x-1?? so confuse

  3. x = 1+y^2
    x-1 = y^2
    √(x-1) = y

    Algebra I, man, Algebra I.

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