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Math Help - Line Integral Problem

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    Line Integral Problem

    Can anyone give me some insight on this problem? I know what it's asking for but I don't have any idea where to start. Thanks!

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

    ∫(y^3-y)i - (2x^3)jĚdr is a maximum.
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    Quote Originally Posted by nickw015 View Post
    Can anyone give me some insight on this problem? I know what it's asking for but I don't have any idea where to start. Thanks!

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

    ∫(y^3-y)i - (2x^3)jĚdr is a maximum.
    For \int_C (y^3-y)dx - 2x^3dy, if we use green theorem then  <br />
\iint_R 1 - 6x^2-3y^2 dA<br />

    If we treat this as a volume problem we are asking for the region R that gives maximum volume. Since z = 1 - 6x^2-3y^2 is an inverted elliptical paraboloid centered at the origin, the region to give then greatest volume is z \ge 0 and thus the boundary of R is 6x^2 + 3y^2 = 1.
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    Another thought w/o Green's thm. If we write J[y] = \int_c L\, dx where L = y^3-y - 2 x^2 y' , then maximizing the functional using the Euler-Lagrange equation will give rise to the same answer.
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