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Lower Level integration problem (Find the centroid)

  1. May 31, 2009 #1
    EDIT - Solved. Thank you.

    1. The problem statement, all variables and given/known data
    Find the Centroid of the region bounded by the x-axis and [tex]\sqrt{9-x^2}[/tex]


    2. Relevant equations
    So far, I found my Mx which ended up being 27p after applying this forumula:
    p[tex]\int[/tex]([tex]\frac{\sqrt{9-x^2}+0}{2}[/tex]) ([tex]\sqrt{9-x^2}[/tex]-0)dx​
    *My interval was [-3,3]

    3. The attempt at a solution

    My problem is when I go to find the mass so I can find the centroid and get my final answer. The equation to find the mass is:

    m=p[tex]\int[/tex][f(x)-g(x)]dx

    so when applying the formula to my problem, I get this:

    m=p[tex]\int[/tex][tex]\sqrt{9-x^2}[/tex]

    So really, it ends up being a simple integration stumble. I just can't figure out how to integrate this. I tried the substitution rule, but it didn't work out.

    Can someone point me in the right direction please?
     
    Last edited: May 31, 2009
  2. jcsd
  3. May 31, 2009 #2
    I'm guessing you have probably gone wrong somewhere unfortunately i don't know where. But the solution to your integral:

    http://www5a.wolframalpha.com/Calculate/MSP/MSP8641961aa713bih43db00000ebbd4gc0a51fd4i?MSPStoreType=image/gif&s=15 [Broken]
     
    Last edited by a moderator: May 4, 2017
  4. May 31, 2009 #3
    Thank you greatly for the reply. I'm in Calc. II, learning all this integration as fast as I can and I stumble here and there in my theorems.

    I was looking at my inverse and hyperbolic trig. functions to try and make a connection, but never thought of double substituting regular sine.

    However, I'm not the kinda person to write an answer/work down and claim it as my own, especially since I won't be able to do it when the test comes, so using your reply I have a few questions.
    • #1 - I notice you didn't see the constant (p) I have to have prior to the integral. I'm guessing that the p would just go along for the ride with the all the other constants.
    • #2 - speaking of constants, how did you get that 9 outside the integral (Step:1/2)? I was thinking you just pulled the 9 out from the integrand; but wouldn't that leave a 1-sin^2?
    Once again, thanks for the responses. I'm going to be lurking on these forums a lot more often now.
     
    Last edited: May 31, 2009
  5. May 31, 2009 #4
    We are using the constant multiple rule for integration and 1-sin^2=cos^2.
     
  6. May 31, 2009 #5
    oh, duh'. I appreciate the help. I'll respond soon w/ my answer.
     
  7. May 31, 2009 #6
    substitute x=3sin(t)..and the answer comes by itself...
     
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