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Line integral across a vector field

  1. Oct 26, 2012 #1
    1. The problem statement, all variables and given/known data
    [tex]\int_C \mathbf F\cdot d \mathbf r[/tex] where [tex]\mathbf F = x^2\vec{i}+e^{\sin^4{y}}\vec{j}[/tex]
    and C is the segment of y=x^2 from (-1,1) to (1,1).

    2. Relevant equations
    [tex]\int_C \mathbf F\cdot d \mathbf r=\int_a^b \mathbf F( \mathbf r(t))\cdot r'(t) dt=\int_C Pdx+Qdy[/tex] where [tex]\mathbf F = P\vec{i}+Q\vec{j}[/tex]

    3. The attempt at a solution
    I parametrize C:
    [tex]\mathbf r(t)=<t,t^2>|-1\leq t \leq 1[/tex]
    Well, I know how to apply the equations above as well as Green's theorem (I use this by letting C2 be a line from (1,1) to (-1,1) and integrating across the enclosed disk D), but no matter what I do, I find myself having to integrate some version of [tex]e^{\sin^4{y}}[/tex]
    There has to be some trick I'm missing. Could someone point me in the right direction?
     
  2. jcsd
  3. Oct 26, 2012 #2

    SammyS

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    If [itex]\displaystyle \mathbf r(t)=<t,\,t^2>\,,\ [/itex] then [itex]\displaystyle d\mathbf r(t)=<1,\,2t>dt\ . [/itex]

    You will have an odd integrand involving the problematic [itex]\displaystyle e^{\sin^4{y}}\ .[/itex]
     
    Last edited: Oct 26, 2012
  4. Oct 26, 2012 #3
    Does this get me anywhere? It seems to still leave me with something unsolveable.
     
  5. Oct 26, 2012 #4

    LCKurtz

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    Check for independence of path and use the straight line from (-1,1) to (1,1).
     
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