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Thread: Line Integrals Across Line Segment

  1. #1
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    Line Integrals Across Line Segment

    Hello all,

    I have a problem where I need to parametrize a line segment in order to evaluate a line integral, but I don't know how to do that. The problem is as follows:

    "Evaluate $\displaystyle \int_C$ $\displaystyle ye^x ds$, where $\displaystyle C$ is the line segment joining $\displaystyle (1,2)$ to $\displaystyle (4,7)$."

    The book has a small example about the parametrization of line segments, but I can't follow it. Can anyone give a blow-by-blow description of how to use the equation: $\displaystyle r(t) = (1-t)r_0 + tr_1 $ where $\displaystyle 0<t<1$?

    I understand that $\displaystyle r_0 $ and $\displaystyle r_1 $ are just the two points, initial and final, but how do you plug them into the previous formula to get $\displaystyle x = [something]$ and $\displaystyle y = [something else]$?

    Thanks for your consideration,

    Austin Martin
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  2. #2
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    Quote Originally Posted by auslmar View Post
    Hello all,

    I have a problem where I need to parametrize a line segment in order to evaluate a line integral, but I don't know how to do that. The problem is as follows:

    "Evaluate $\displaystyle \int_C$ $\displaystyle ye^x ds$, where $\displaystyle C$ is the line segment joining $\displaystyle (1,2)$ to $\displaystyle (4,7)$."

    The book has a small example about the parametrization of line segments, but I can't follow it. Can anyone give a blow-by-blow description of how to use the equation: $\displaystyle r(t) = (1-t)r_0 + tr_1 $ where $\displaystyle 0<t<1$?

    I understand that $\displaystyle r_0 $ and $\displaystyle r_1 $ are just the two points, initial and final, but how do you plug them into the previous formula to get $\displaystyle x = [something]$ and $\displaystyle y = [something else]$?

    Thanks for your consideration,

    Austin Martin
    Hi

    This is how I would have performed the job

    One equation of the line segment joining $\displaystyle (1,2)$ to $\displaystyle (4,7)$ is 5x-3y+1=0 or y=5x/3 + 1/3

    You can deduce that from one point of the line (x,y) to a very close one (x+dx,y+dy) you have the relationship dy=5dx/3

    Now $\displaystyle ds = \sqrt{dx^2+dy^2}=\sqrt{dx^2+\frac{25}{9}dx^2}=\fra c{\sqrt{34}}{3}dx$

    Therefore integral becomes

    $\displaystyle \int_C ye^x ds = \int_{1}^{4} (\frac{5}{3}x+\frac{1}{3})e^x\frac{\sqrt{34}}{3}dx$

    .. but this method does not match with the parametrization you are looking for
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