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Homework Help: Integral Problem (very Hard)

  1. Mar 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that [tex]\int[/tex][tex]\sqrt{}9-x^2[/tex] dx


    given that x=3sin[tex]\Theta[/tex]
    2. Relevant equations

    3. The attempt at a solution
    [tex]\int[/tex][tex]\sqrt{}9-x^2[/tex] dx





    Im not to sure if im going in the right direction if i am not guidance would be appreciated
  2. jcsd
  3. Mar 4, 2010 #2


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    You're not going in the right direction. Your very first step is wrong. The power rule only applies when the integrand is of the form xndx where [itex]n \ne -1[/itex]. It doesn't apply when you have some function of x taken to a power, as you do in this case. Also, I have no idea where that 10x in the denominator came from.

    Use the substitution given and rewrite the integral in terms of [itex]\theta[/itex] first.
  4. Mar 4, 2010 #3
    ok so if i do...


    is it going in the right direction now?? thanks for the help
    Last edited: Mar 4, 2010
  5. Mar 4, 2010 #4


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    Marginally better. First of all,

    [tex]\sqrt{a^2-b^2} \ne a-b[/tex]

    Second, you forgot the dx and then didn't write it in terms of [itex]d\theta[/itex]. Finally, not that it really matters, you didn't integrate the first term correctly.

    I would suggest you review your textbook on the topic of trig substitutions. There's probably a similar example you could use as a template for solving this problem.
  6. Mar 4, 2010 #5


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    [itex]\sqrt{9- 9 sin^2(\theta)}= 3\sqrt{1- sin^2(\theta)}= 3\sqrt{cos^2(\theta)}[/itex]
  7. Mar 4, 2010 #6
    okay i was finally able to prove it!!

    [tex]\int\sqrt{a^2-x^2}[/tex] dx
    =[tex]\int a^2-a^2sin^2\Theta[/tex] acos[tex]\Theta[/tex] d[tex]\Theta[/tex]
    =[tex]\int\sqrt{a^2(1-sin^2\Theta}[/tex] acos[tex]\Theta[/tex] d[tex]\Theta[/tex]
    =[tex]\int\sqrt{a^2cos^2\Theta}[/tex] acos[tex]\Theta[/tex] d[tex]\Theta[/tex]
    =[tex]\int a^2cos^2\Theta[/tex] d[tex]\Theta[/tex]


    i know i definatly skipped a couple of steps in integrating cos^2 :redface:
    Last edited: Mar 4, 2010
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