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Green's Theorem & Line Integral confusion

  1. May 18, 2012 #1
    1. The problem statement, all variables and given/known data
    a) Evaluate the work done by the force field F(x, y) = (3y^(2) + x)i + 4x^(3)j over the curve
    r(t) = e^(t)i + e^(3t)j, tε[0, ln(2)].
    b) Using Green’s theorem, find the area enclosed by the curve r(t) and the segment that
    joins the points (1, 1) and (2, 8).
    c) Find the flux of F across the curve described in b).


    2. Relevant equations
    I may be missing something but for the life of me I can't figure out how to answer part b.). I already have part a.) and can do part c.) just need to figure out the limits for part be.


    3. The attempt at a solution
    a.)∫(0 to ln2 )[3e^(7t) + e^(2t) + 12e^(7t)]dt= 2547

    B.)Greens Theorem
    ∫(12x^2-(6y+1))dA

    Any help would be awesome!
     
  2. jcsd
  3. May 18, 2012 #2

    sharks

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    Gold Member

    For part (b): Draw the graph of the curve r(t) in the given interval. You only need to plot 3 points to get a general idea of the shape of the graph. Try the following values of t: 0, ln (1) and ln (2). Then, plot the line that joins the points (1,1) and (2,8). Find its equation. Then describe the enclosed region and find its area using the Green's theorem.
     
  4. May 18, 2012 #3

    tiny-tim

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    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi Dead85! Welcome to PF! :smile:

    (try using the X2 button just above the Reply box :wink:)
    So r is part of y = x3.

    To find the area, you need ∫∫ 1 dA.

    So to use Green's theorem, you need a function (G(x,y),H(x,y)) with ∂H/∂x - ∂G/∂y = 1.

    Try something like (0,-x) or (y,0). :smile:
     
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