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Problem 12 Easy Difficulty

Use Green's Theorem to evaluate $ \displaystyle \int_C \textbf{F} \cdot d\textbf{r} $. (Check the orientation of the curve before applying the theorem.)

$ \textbf{F}(x, y) = \langle e^{-x} + y^2, e^{-y} + x^2 \rangle $,
$ C $ consists of the arc of the curve $ y = \cos x $ from $ (-\pi/2, 0) $ to $ (\pi/2, 0) $ and the line segment from $ (\pi/2, 0) $ to $ (-\pi/2, 0) $

Answer

$\frac{\pi}{2}$

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Video Transcript

So in this problem, we want to start by looking at the partial derivative of Q. With respect to X and the partial derivative of P. With respect to y eso, we recognize that the partial derivative with respect the partial derivative of Q with respect to X will be to why, for the Q. With respect to X, it'll be two X and then two y. So what we end up getting as a result is ultimately that we'll have the integral from negative high over to however, to okay of the integral from zero to co sign of acts of two y minus two x. Why the X then we first want to focus on this center role on when we do. We see that this is going to eventually give us a cosine squared X onda simplifying out the coastline squared X We end up seeing that this will give us pi over two. What do you see is the same value is right here. So with that, we know that power to is our answer. Using green serum green serum is really helpful because rather than having to deal with, um, ese and exponents and all that were able to simplify things much easier into just two y minus two x. So that's why Green Thumb really comes in handy in this case.

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