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The (im)Possible Integral

  1. Jun 21, 2011 #1
    1. The problem statement, all variables and given/known data

    Evaluate the indefinite integral.

    [itex]\int\sqrt{28x-x^2} dx[/itex]

    3. The attempt at a solution

    [itex]\int\sqrt{196-(x-14)^2} dx[/itex]
    Completing the square​

    [itex]u=x-14[/itex]

    [itex]du=dx[/itex]

    [itex]\int\sqrt{196-u^2} du[/itex]
    u substitution​

    [itex]u=14sin\theta[/itex]

    [itex]du=14cos\theta d\theta[/itex]
    Trig substitution​

    [itex]\int\sqrt{196cos^2\theta} 14cos\theta d\theta[/itex]

    [itex]\int14cos\theta*14cos\theta d\theta[/itex]

    [itex]98\int1+cos2\theta d\theta[/itex]

    [itex]98(\theta+sin\theta*cos\theta) + C[/itex]

    [itex]98(arcsin(u/14)+(u/14)(\sqrt{196-u^2}/14)[/itex]
    Solve for Theta​

    From here I should be able to just sub in for u and arrive at my answer. Unfortunately, it is close but no correct, and I can't see where I'm going wrong. Any ideas?
     
  2. jcsd
  3. Jun 21, 2011 #2
    What's the answer you're getting and what's the answer you're supposed to get. I don't see any errors in what you've written, but perhaps its a simplification error.
     
  4. Jun 21, 2011 #3

    micromass

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  5. Jun 21, 2011 #4
    Thanks for the prompt replies :)

    So it appears my problem is either data entry or substitution; that's a good sign. Here is my final solution exactly as I've entered it in our software:

    98(arcsin((x-14)/14)+(x-14)sqrt(x^2-28x)/2)

    Sorry if that's not to pretty; here's the image of the same input:

    [PLAIN]http://webwork.asu.edu/webwork2_files/tmp/equations/a6/bd057be8bc635789c21be6e19460f51.png [Broken]

    Comparing it to the Wolfram output, it looks close but not identical. Then again, the computer tends to do some crazy simplification at the end; maybe they're equivalent. See: http://www.wolframalpha.com/input/?i=integrate+sqrt(28x-x^2)
     
    Last edited by a moderator: May 5, 2017
  6. Jun 21, 2011 #5

    micromass

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    Doesn't that last term need to be

    [tex]\frac{(x-14)\sqrt{28x-x^2}}{196}[/tex]

    So you need to switch the entries in your root around. and I don't really see why you only divide by 2.

    The wolfram output is correct. However, wolfram uses an entirely different algorithm to calculate this. So the answers can look pretty different...
     
    Last edited by a moderator: May 5, 2017
  7. Jun 21, 2011 #6
    You're right; I see my error now. On paper, I distributed the 98*stuff/196 and got stuff/2, but when I entered it I kept the 98 as a factor. Edit: And you're right about the order too- I reversed it. Always rushing these things at the end.

    Let me try that and see what happens. Thanks again!
     
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