Let F be the vector field F=(13x^2 y+3y^3 -y)i -12x^3 j. Find the maximum value of
∫F•dr where c is positively oriented simple closed curve.

Question

Let F be the vector field F=(13x^2 y+3y^3 -y)i -12x^3 j. Find the maximum value of
∫F•dr where c is positively oriented simple closed curve.

ganeshie8 · Answer

You may start with green's thm I think

ganeshie8 · Answer

@dan815

ganeshie8 · Answer

$$\oint F.dr = \iint\limits_{R} \text{curl}(F)~dA =  \iint\limits_{R}  -49x^2-9y^2+1~dA $$

ganeshie8 · Answer

Easy to see that the integrand stays positive in the region $49x^2+9y^2 \lt 1$, so should we pick this as region of integration ?

frank0520 · Answer

Here is a picture of the question.

ganeshie8 · Answer

Then we're correct, go ahead and evaluate the integral over that ellipse

ganeshie8 · Answer

$$max(\oint F.dr) ~= ~\iint\limits_{49x^2+9y^2\lt 1}~~1-49x^2-9y^2~dA$$

ganeshie8 · Answer

you will need to use change of variables

ganeshie8 · Answer

frank0520 · Answer

looks correct to me, I got $$\frac{ 42\pi }{ 4 }$$