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Math Help - Compute the line integral

  1. #1
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    Compute the line integral

    Calculate line integral \int_c xydx + x^2y^3dy, where c is triangle with vertices at points (0,0), (1,0) and (1,2).

    How can I find r(t) ?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Apprentice123 View Post
    Calculate line integral \int_c xydx + x^2y^3dy, where c is triangle with vertices at points (0,0), (1,0) and (1,2).

    How can I find r(t) ?
    you have to state what direction you are going in. is it from (0,0) to (1,0) to (1,2)??

    you will have to do three separate integrals, one for each side of the triangle. for each, your r(t) will be the vector function for the line.

    Here is how to proceed:

    Assuming we are going from (0,0) to (1,0) to (1,2), call the line connecting (0,0) to (1,0) C_1, the line connecting (1,0) and (1,2) C_2 and the line connecting (1,2) back to (0,0) C_3.

    Then we have \int_C F = \int_{C_1}F + \int_{C_2}F + \int_{C_3}F.

    lets concentrate on C_1.

    for C_1, we have the line y = 0, with x ranging from 0 to 1. thus, we can parametrize the line by:

    x = t, y = 0, for 0 \le t \le 1

    and so, dx = 1, dy = 0, and our integral for C_1 becomes:

    \int_0^1 t(0)(1) + t^2(0)^3(0)~dt = \int_0^1 0~dt = 0


    now do the same for C_2 and C_3.


    (note, there is no need to introduce a t here. we could have parametrized our line by x = x, y = 0 for 0 \le x \le 1)
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  3. #3
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    Line integral

    thank you
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