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Homework Help: Approximating PI

  1. Sep 29, 2009 #1
    Approximating PI Show that [tex]\int[/tex][tex]\stackrel{1}{0}[/tex][tex]\frac{x^{4}(1-x)^{4}}{1+x^{2}}[/tex]dx=[tex]\frac{22}{7}[/tex]-[tex]\Pi[/tex] Why does this imply that [tex]\Pi[/tex][tex]\triangleleft[/tex][tex]\frac{22}{7}[/tex]

    I have no clue where to begin with this, I'm at a loss, this is one of the questions for in my university project, first year. Any help is appreciated.
  2. jcsd
  3. Sep 29, 2009 #2
    Calculating the integral would probably be a good start.

    What's the numerical value of the integral? Is it a big number or a small number? Is it positive or negative?
  4. Sep 30, 2009 #3
    Welcome to PF, Zadey.

    To evaluate [tex]\int_0^1 \frac{x^{4}(1-x)^{4}}{1+x^{2}}\,dx[/tex]

    multiply out the numerator, then use long division, then integrate from 0 to 1.
  5. Sep 30, 2009 #4


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    Once you have done the integral and derived the result shown, if [itex]\pi[/itex] were greater than 22/7, the integral would be negative.
  6. Sep 30, 2009 #5
    Thanks, I got it. Don't know why I didin't see it earlier.
    Now if I had to find the maximum of the numerator, how would I go about using it to show that [tex]\frac{22}{7}[/tex]<[tex]\frac{\pi}{1024}[/tex]<[tex]\frac{1}{100}[/tex] and how does it imply that the approximation [tex]\frac{22}{7}[/tex]is accurate to 2 decimal places? I know that the [tex]\frac{1}{100}[/tex] would be used to imply that its accurate to 2 decimal places but how it does I'm not sure.
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