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Question 18: Green's TheoremProctorLet C be the curve in the plane with parametrization r(t) (3 cos(t), -3 sin(t)) , where t € [0, 27]. Let F(c,y) = (2y2...

Question

Question 18: Green's TheoremProctorLet C be the curve in the plane with parametrization r(t) (3 cos(t), -3 sin(t)) , where t € [0, 27]. Let F(c,y) = (2y2 9, 4xy + 2x) . Compute the line integral Jc F dr and enter your answer below:JcF e drnumber (3 significant figures)

Question 18: Green's Theorem Proctor Let C be the curve in the plane with parametrization r(t) (3 cos(t), -3 sin(t)) , where t € [0, 27]. Let F(c,y) = (2y2 9, 4xy + 2x) . Compute the line integral Jc F dr and enter your answer below: JcF e dr number (3 significant figures)



Answers

Apply Green's Theorem to evaluate the integrals in Exercises $27-30 .$ $$\oint\left(2 x+y^{2}\right) d x+(2 x y+3 y) d y$$ C: Any simple closed curve in the plane for which Green's Theorem holds

The problem. Well, the giver region is a square triangle Before we can apply the green theory, Um, we say that the integration off course in X sign x the x co sign X sign Why India plus sine x co sign Why in right where this is from X, right? And this is your backs. Why? So this can be written at a level integration co sign X crow Sign y minus Goldstein X call sign. Why d why the X, which is equal toe you?

A problem. Number five sense. In our case, f of X y is equal to x co sign why n g of X y is equal to negative. Why sine x and the square with Vertex zero and zero by and zero by over two and by over two and zero and pi over two. Satisfy the hypothesis off the green theory. Um, so we have the integration off X course I And why the X minus? Why Sign X? Yeah, I which is equal to the integration off X sign. Why mine? That's why Goldstein x e y de yet which is equal to ah five square over eight minus by squared over eight So is able to do.

Alright problem 1 80 Using greens here and we're gonna evaluate the integral. Okay, then involving this integral, we see that it comes out to in the door of zero D A, which is a good 20 Yeah.


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