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Thread: Find the positive simple closed surface for which the line integral is the max?

  1. #1

    (UPDATED) Maximize a double integral

    Find the positively oriented simple closed surface C, for which the value of the line integral

    Sc ((y^3)-y)dx - 2x^3dy S is the elongated S-integral symbol

    is a maximum.

    I was having trouble figuring this out. I was wondering if I have to go back to a possible original function to do this.

    I know that Sc Pdx+Qdy = Sc ((y^3)-y)dx - 2x^3dx So P=(y^3)-y and Q=2x^3. So then F(x,y)=[(y^3)-y]i + [2x^3]j.

    Our teacher said: Hint: First use Green's Theorem. Consider the resulting double integral. How would you maximize it?

    So, using Green's Theorem of
    Sc Pdx+Qdy= SS(Qx-Py)dA

    The partial derivatives are
    Qx= 6x2 and Py= 3y2-1

    SS(over the domain) (6x2-3y2+1)dA

    Now according to the teacher, how would you maximize it?

    I am not sure how to maximize it because I am thinking it would go to infinity? Any help? Thanks!
    Last edited by crybloodwing; 12-06-2018 at 10:26 AM.


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