# Math Help - Line Integral Problem

1. ## 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)j·dr is a maximum.

2. Originally Posted by nickw015
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)j·dr is a maximum.
For $\int_C (y^3-y)dx - 2x^3dy$, if we use green theorem then $
\iint_R 1 - 6x^2-3y^2 dA
$

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 $6x^2 + 3y^2 = 1$.

3. 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.