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Math Help - independence of path and evaluating integral

  1. #1
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    independence of path and evaluating integral

    Hi,
    I have to test for independence of path and evaluate the integral.
    I could prove that it's independence of path but I couldnt evaluate the integral.

    (3,4)
    ∫ (2x+5y-7)dx + (5x-8y+3)dy
    (1,1)

    where C: 3x - 2y-1=0
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  2. #2
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    Quote Originally Posted by panda View Post
    Hi,
    I have to test for independence of path and evaluate the integral.
    I could prove that it's independence of path but I couldnt evaluate the integral.

    (3,4)
    ∫ (2x+5y-7)dx + (5x-8y+3)dy
    (1,1)

    where C: 3x - 2y-1=0
    Say that \bold{f}(x,y) = (2x+5y-7,5x-8y+3).
    We want to find F(x,y) so that \nabla F = \bold{f}.
    This mean \partial_1 F = 2x+5y - 7 \text{ and }\partial_2 F = 5x - 8y+3.
    The first equation becomes F(x,y) = x^2 + 5xy - 7x + g(y).
    But then \partial_2 F (x,y) = 5x + g'(y).
    Thus we want g'(y) = -8y + 3 the function g(y) = -4y^2 + 3y works.
    Thus, an example of a function is F(x,y) = x^2 + 5xy -7x - 4y^2 + 3y.

    Then the integral is F(3,4) - F(1,1) - compute that for the answer.
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