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


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Aug 22, 2017
(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!
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