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Math Help - Line Integrals

  1. #1
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    Line Integrals

    Evaluate the line integral  \int_C sin \ x \ dx + cos \ y \ dy where C consists of the top half of the circle x^2+y^2=1 from (1,0) to (-1,0) and the line segment from (-1,0) to (-2,3)

    I got 0 for the line integral over the circle, and Cos \ 1- Cos \ 2 +Sin \ 3 over the line segment. However, the answer provided is Cos \ 1- Cos \ 2 +Cos \ 3. Can someone check?

    Also, anyone know a way to evaluate line integrals on Mathematica?
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  2. #2
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    Your line integral is right, I just checked. Maybe the arc needs to be expressed solely in terms of x or y, rather than x^2 + y^2 = 1? For example replace y with sqrt(1 - x^2). However the integral becomes very complicated after that..
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  3. #3
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    Thanks, I'll ask my TA if she possibly made a mistake.
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  4. #4
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    Quote Originally Posted by Andrew007 View Post
    Your line integral is right, I just checked. Maybe the arc needs to be expressed solely in terms of x or y, rather than x^2 + y^2 = 1? For example replace y with sqrt(1 - x^2). However the integral becomes very complicated after that..
    Quick question though: would such a substitution change anything? It's still the same expression.
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  5. #5
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    Quote Originally Posted by MathTooHard View Post

    Also, anyone know a way to evaluate line integrals on Mathematica?
    Mathematica can do line integrals if you supply the parameterization:

    Here's yours:

    Code:
    In[24]:=
    x[t_] := Cos[t]; 
    y[t_] := Sin[t]; 
    Integrate[Sin[x[t]]*Derivative[1][x][t], 
       {t, 0, Pi}] + Integrate[
       Cos[y[t]]*Derivative[1][y][t], 
       {t, 0, Pi}]
    x[t_] := t; 
    y[t_] := -3*t - 3; 
    Integrate[Sin[x[t]]*Derivative[1][x][t], 
       {t, -1, -2}] + Integrate[
       Cos[y[t]]*Derivative[1][y][t], 
       {t, -1, -2}]
    
    Out[26]=
    0
    
    Out[29]=
    Cos[1] - Cos[2] + Sin[3]
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