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Math Help - Integrals

  1. #1
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    Integrals

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    Last edited by mamk; January 16th 2009 at 11:58 AM. Reason: wrong post
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by mamk View Post
    Where's the question?
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  3. #3
    MHF Contributor

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    Quote Originally Posted by mamk View Post
    The problem is
    1) Show that the line integral \int ye^{xy}dx+ xe^{xy}dy is independent of the path.
    2) Find the integral from (1, 0) to (0, 4)
    A line integral such as that is independent of the path if and only if the differential being integrated is an "exact differential"- that is, that there exist a function F(x,y) such that dF= F_xdx+ F_ydy= ye^{xy}dx+ (xe^{xy}- 2y)dy. Since we must have [tex]F_{xy}= F_{yx}, that, in turn, is true if and only if (ye^{xy})_y= (xe^{xy}- 2y)_x. Is that true?

    Once you have shown that, you can do (2) by actually finding that F(x,y).
    We must have F_x= ye^{xy}. Integrating that, treating y as a constant, F(x,y)= e^{xy}+ g(y). (Since we are treating y as a constant, the "constant of integration" may be a function of y- that is the "g(y)".)

    From F(x,y)= e^{xy}+ g(y) and so F_y= xe^{xy}+ g'(y)= ye^{xy}-2y. The " ye^{xy}" cancels out- that had to happen in order that g' be a function of of y only and happens precisely because of the condition in (1). That tells us g'(y)= -2y so that g(y)= -y^2+ C where C now really is a constant. That is, F(x,y)= e^{xy}- 2y. Evaluate that at (0,4) and (1, 4) and subtract.
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