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Homework Help: Volume of a solid w/known cross section

  1. Oct 5, 2011 #1
    1. The problem statement, all variables and given/known data

    The base region of a solid is bounded by y=x, y=(x-1)^2, and x = 1.

    The cross sections are semicircles perpendicular to the x-axis.

    Write a riemann sum and definite integral.

    2. Relevant equations

    3. The attempt at a solution

    First, I wrote down the formula for a semicircular disk's volume. 1/2(pi(r^2)(h))

    I then found the intersection of y=x and y=(x-1)^2 to be .382 and another value that was greater than 1, so I ditched it.

    I then wrote down the diameter of any given disk as x - (x-1)^2 or -x^2 + 3x - 3, so radius is half of that, and I defined the height of each disk to be delta x.

    So, I wrote the Riemann Sum as: (limit as delta x approaches 0)

    [itex]\Sigma \frac{\pi}{4}(-x^{2}+3x-3)^{2}\Delta x[/itex]

    And therefore wrote a definite integral as:

    [itex]\frac{\pi}{4} \int^{1}_{.382} (-x^{2}+3x-3)^{2} dx[/itex]

    Did I do this right?
  2. jcsd
  3. Oct 5, 2011 #2


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    Staff Emeritus
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    It looks good. The true value for the lower limit of integration is, [itex]\displaystyle \frac{3-\sqrt{5}}{2}\,.[/itex]
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