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Homework Help: Area Between Curves

  1. Mar 21, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the area of the region between the two curves.

    [tex]y=\sqrt{x}[/tex]

    [tex]y=\frac{1}{2}x[/tex]

    [tex]x=9[/tex]

    2. Relevant equations



    3. The attempt at a solution

    The domain of the region is [4,9]:

    [tex]\int\frac{1}{2}x-\sqrt{x}dx[/tex] with limits of integration [tex][4, 9][/tex]

    [tex]=\frac{1}{2}\intxdx-\int x^\frac{1}{2}dx[/tex]

    [tex]=\frac{1}{2}\frac{x^2}{2}-\frac{2}{3}x^\frac{3}{2}[/tex]

    [tex]=\frac{1}{4}(9^2-4^2)-\frac{2}{3}(9^\frac{3}{2}-4^\frac{3}{2})[/tex]

    [tex]=\frac{1}{4}(65)-\frac{2}{3}(27-8)[/tex]

    [tex]=\frac{1}{4}(65)-\frac{2}{3}(19)[/tex]

    [tex]=\frac{65}{4}-\frac{38}{3}[/tex]

    [tex]=\frac{195-152}{12}[/tex]

    [tex]=\frac{43}{12}[/tex]

    The answer in the book is [tex]\frac{59}{12}[/tex].
     
  2. jcsd
  3. Mar 21, 2010 #2

    Dick

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    Science Advisor
    Homework Helper

    It looks to me like there are two parts to the region lying between the two curves. What about the [0,4] part? Shouldn't you add the areas of both of them?
     
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