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Math Help - Fundamental Theorem of Line Integrals

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    Fundamental Theorem of Line Integrals

    How do I use the fundamental theorem of line integrals to evaluate the integral
    (3x-y+1)dx - (x+4y+2)dy from (-1,2) to (0,1)?
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    Quote Originally Posted by noles2188 View Post
    How do I use the fundamental theorem of line integrals to evaluate the integral
    (3x-y+1)dx - (x+4y+2)dy from (-1,2) to (0,1)?
    first find the potential function (do you know how to do this?), call it f(x,y). then your line integral is equal to f(-1,2) - f(0,1)
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    that's the part i am having trouble with, finding f.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by noles2188 View Post
    that's the part i am having trouble with, finding f.
    *sigh* i know i responded to similar questions here, but i can only find the ones with 3 variables, not two, so they might confuse you. i can't find a way to give you hints for this other than do the problem...or most of it anyway. other than that, i will tell you to go read your textbook. i know they gave an example of how to do this

    Quote Originally Posted by noles2188 View Post
    How do I use the fundamental theorem of line integrals to evaluate the integral
    (3x-y+1)dx - (x+4y+2)dy from (-1,2) to (0,1)?
    we want to find a function f(x,y) so that f_x = 3x - y + 1 and f_y = -x - 4y - 2

    (we know we can do this since the "equations" are exact: \frac {\partial }{\partial y}(3x - y + 1) = \frac {\partial }{\partial x}(-x - 4y - 2))

    so take f_x = 3x - y + 1, then

    f(x,y) = \int f_x~dx = \frac 32x^2 - xy + x + g(y)

    (our constant of integration is a function of y, since this will die when we differentiate with respect to x)

    \Rightarrow f_y = \frac {\partial }{\partial y}f(x,y) = -x + g'(y)

    But f_y = -x - 4y - 2, so we have -x -4y - 2 = -x + g'(y). hence

    g'(y) = -4y - 2

    \Rightarrow g(y) = -2y^2 - 2y + K


    Therefore, f(x,y) = \frac 32x^2 - xy + x -2y^2 - 2y + K is the (potential) function we seek
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    they do show an example of this in my book but they leave out some steps so i was a little bit confused, but now i think i have decent understanding of it. thanks.
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