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Math Help - Line Integral over a Plane Curve

  1. #1
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    Line Integral over a Plane Curve

    Okay I'm having trouble parameterizing the curve C: x^2 + y^2 = 4 from (0,2) to (SQRT 2, SQRT 2) in the first quadrant in order to take the line integral for f (x,y) = x^2-y

    I understand to put C into polar coordinates, and that r=2 but my key shows that r(t) = (2cost)i + (2sint)j 0<=t=<pi/2

    and I have no idea how to get that r(t)...
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Typically for a circle oriented counter clockwise you use

    x= rcos(t) y = rsin(t) starting at (r,0)


    Here though you are moving clockwise from(0,2) so switch the sine and cosine

    use x = 2sin(t)

    y= 2cos(t) and let t vary from 0 to pi/4
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  3. #3
    MHF Contributor Calculus26's Avatar
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    By the way

    x= 2cos(t) y =2sin(t) 0<t<pi/2

    is a counterclockwise parameterixation starting at (2,0) aand going to (0,2)
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  4. #4
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    I see so any time you go counterclockwise as far as polar coordinates go, the x and y are flipped then?
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  5. #5
    MHF Contributor Calculus26's Avatar
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    you mean clockwise? It depends on where you start

    Eg if you start at (-1,0) trvrling clockwise then x(t) =-cos(t) y(t) =sin(t)
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