Question: Let (I=oint_{C} mathbf{F} cdot d mathbf{r}), where (mathbf{F}(x, y)=leftlangle y+sin x^{2}, x^{2}+e^{y^{2}}ightangle) and (C) is the circle of radius 4 centered at the origin. (a)

Let $I=\oint_{C} \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y)=\left\langle y+\sin x^{2}, x^{2}+e^{y^{2}}ightangle$ and $C$ is the circle of radius 4 centered at the origin.
(a) Which is easier, evaluating $I$ directly or using Green's Theorem?
THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D

(b) Evaluate $I$ using the easier method.

THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D is a simple closed curve, oriented counterclockwise. If F and F have continuous partial deriva- tives in an open region containing D, then $ F1 dx + F dy aF2 = -IL (F 3F) A ax - 2

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