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Math Help - Find the work done by F in moving a body counterclockwise around a curve

  1. #1
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    Find the work done by F in moving a body counterclockwise around a curve

    F=(x^{2}+y^{3})i+(2xy)j C:x^{2}+y^{2}=9

    I'm not totally sure how to do this problem. So far, I've done this:

    r(t)=rcos(t)i+rsin(t)j => 3cos(t)i+3sin(t)j 0\leq t \leq 2\pi

    F=(9cos^{2}(t)+27sin^{3}(t))i+18sin(t)cos(t)j

    \frac{dr}{dt}=-3sin(t)i+3cos(t)j

    F * \frac{dr}{dt}== -3sin(t)(9cos^{2}(t)+27sin^{3}(t))+3cos(t)(18sin(t)  cos(t))

    => -27sin(t)cos^{2}(t)+81sin^{4}+48sin(t)cos^{2}(t)

    \displaystyle \int^{2\pi}_{0}81sin^{4}+21sin(t)cos^{2}(t)dt

    From here, I don't know how to calculate the integral. Is there a trig identity that simplifies the problem? Do I have to integral by parts or substitution? I plugged it into wolframalpa and got a solution but it didn't give me what I really wanted, the steps on how to solve the integral.

    Am I even approaching the problem correctly? Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by downthesun01 View Post
    F=(x^{2}+y^{3})i+(2xy)j C:x^{2}+y^{2}=9
    \displaystyle \int^{2\pi}_{0}81sin^{4}+21sin(t)cos^{2}(t)dt From here, I don't know how to calculate the integral.
    For the first one: \sin ^4t=[(1/2)(1-\cos 2t]^2=\ldots. For the second one: v=\cos t.

    Am I even approaching the problem correctly? Thanks
    Yes, you are.

    Regards.
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    Thanks for the help. I'm not getting the tips for integration that you gave. For the first one, is that a trig substitution? And for the second one, am I supposed to used integration by parts or is that a substitution? I'm really terrible at integrating trig functions.

    Thank you, I've computed the integrals now.
    Last edited by downthesun01; November 16th 2010 at 11:12 PM.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by downthesun01 View Post
    For the first one, is that a trig substitution?
    No, it isn't. We use trigonometric formulas.

    And for the second one, am I supposed to used integration by parts or is that a substitution? I'm really terrible at integrating trig functions.
    It is a substitution:

    \int \sin t\cos^2tdt=-\int v^2dv=-v^3/3+C=-\cos^3t/3+C

    Regards.
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