- #1
usn7564
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Homework Statement
Excuse my terminology, not sure what the actual translations are.
Find a simple (no holes in it), closed, positively oriented, continuously differentiable curve T in the plane such that:
[tex]\int_{T}(4y^3+y^2x-4y)dx + (8x +x^2y-x^3)dy[/tex]
is as big as possible, finally calculate the value of the integral.
The attempt at a solution
I used Green's Theorem to get a rather simple (or so I figured) expression:
[tex]12\iint_{D}1-(\frac{x^2}{4}+y^2)dxdy[/tex]
This made it clear (I presume) that I wanted the ellipse satisfying the equation [tex]1 \leq \frac{x^2}{4}+y^2[/tex]
Then I parametrised it with
[tex] x = 2cos\theta [/tex]
[tex] y = sin\theta[/tex]
And got
[tex]12 \cdot 2\iint r(1-r^3)drd\theta[/tex]
With the 2 and extra r due to the Jacobian determinant. Solved it with [tex]0 \leq \theta \pi[/tex] and [tex]0 \leq r \leq 1[/tex].
I get 6pi, done it numerous times and it stays at 6pi. The answer should be 12pi. Any idea where I'm messing up? I haven't really parametrized an ellipse for a while but I don't remember there being any oddities there, though I guess that's an area where I could have messed up. Quite google search lead me to believe it's not it though, so really not sure.