Remove three incorrect answers from the equation. Always be aware of your errors. Pay attention to the connotation and context of what you're saying. Use Make the most of the evidence questions. Read passages in a strategic manner. Passage Introductions Should Not Be Ignored. Become engrossed in the passages. Before taking the test, read the section instructions. First, answer the questions you already know the answers to. Remove any wrong responses. Maintain a tidy appearance. Keep an eye out for stray marks. The majority of the time, your initial response is correct. There is just one response that is correct.
Answer;
B. Dan una caminata por el sendero.
Explanation:
The preposition POR in Spanish can have a lot of different meanings depending on the context. I
In the B. sentence, por means through.
Dar una caminata POR el sendero. = To take a walk THROUGH the path.lanation:
Given that y = cos(x) makes up part of the boundary of C, I suspect you mean the given points to be (-π/2, 0) and (π/2, 0).
I also assume the given vector field is
![\vec F(x,y) = \left\langle e^{-x} + y^2, e^{-y} + x^2 \right\rangle](https://tex.z-dn.net/?f=%5Cvec%20F%28x%2Cy%29%20%3D%20%5Cleft%5Clangle%20e%5E%7B-x%7D%20%2B%20y%5E2%2C%20e%5E%7B-y%7D%20%2B%20x%5E2%20%5Cright%5Crangle)
Since
has no singularities on C or in its interior, Green's theorem applies:
![\displaystyle \int_C \vec F(x,y) \cdot d\vec r = \iiint_D \frac{\partial(e^{-y}+x^2)}{\partial x} - \frac{\partial(e^{-x}+y^2)}{\partial y} \, dA = 2 \iiint_D (x + y) \, dA](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_C%20%5Cvec%20F%28x%2Cy%29%20%5Ccdot%20d%5Cvec%20r%20%3D%20%5Ciiint_D%20%5Cfrac%7B%5Cpartial%28e%5E%7B-y%7D%2Bx%5E2%29%7D%7B%5Cpartial%20x%7D%20-%20%5Cfrac%7B%5Cpartial%28e%5E%7B-x%7D%2By%5E2%29%7D%7B%5Cpartial%20y%7D%20%5C%2C%20dA%20%3D%202%20%5Ciiint_D%20%28x%20%2B%20y%29%20%5C%2C%20dA)
where D is the interior of C, the region
![D = \left\{ (x, y) : -\dfrac\pi2 \le x \le \dfrac\pi2 \text{ and } 0 \le y \le \cos(x) \right\}](https://tex.z-dn.net/?f=D%20%3D%20%5Cleft%5C%7B%20%28x%2C%20y%29%20%3A%20-%5Cdfrac%5Cpi2%20%5Cle%20x%20%5Cle%20%5Cdfrac%5Cpi2%20%5Ctext%7B%20and%20%7D%200%20%5Cle%20y%20%5Cle%20%5Ccos%28x%29%20%5Cright%5C%7D)
The integral then reduces to
![\displaystyle 2 \iint_D (x + y) \, dA = 2 \int_{-\frac\pi2}^{\frac\pi2} \int_0^{\cos(x)} (x + y) \, dy \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%5Ciint_D%20%28x%20%2B%20y%29%20%5C%2C%20dA%20%3D%202%20%5Cint_%7B-%5Cfrac%5Cpi2%7D%5E%7B%5Cfrac%5Cpi2%7D%20%5Cint_0%5E%7B%5Ccos%28x%29%7D%20%28x%20%2B%20y%29%20%5C%2C%20dy%20%5C%2C%20dx)
![\displaystyle 2 \iint_D (x + y) \, dA = 2 \int_{-\frac\pi2}^{\frac\pi2} \left( x\cos(x) + \frac12 \cos^2(x) \right) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%5Ciint_D%20%28x%20%2B%20y%29%20%5C%2C%20dA%20%3D%202%20%5Cint_%7B-%5Cfrac%5Cpi2%7D%5E%7B%5Cfrac%5Cpi2%7D%20%5Cleft%28%20x%5Ccos%28x%29%20%2B%20%5Cfrac12%20%5Ccos%5E2%28x%29%20%5Cright%29%20%5C%2C%20dx)
![\displaystyle 2 \iint_D (x + y) \, dA = 2 \int_{-\frac\pi2}^{\frac\pi2} \left( x\cos(x) + \frac{1 + \cos(2x)}4 \right) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%5Ciint_D%20%28x%20%2B%20y%29%20%5C%2C%20dA%20%3D%202%20%5Cint_%7B-%5Cfrac%5Cpi2%7D%5E%7B%5Cfrac%5Cpi2%7D%20%5Cleft%28%20x%5Ccos%28x%29%20%2B%20%5Cfrac%7B1%20%2B%20%5Ccos%282x%29%7D4%20%5Cright%29%20%5C%2C%20dx)
![\displaystyle 2 \iint_D (x + y) \, dA = \frac12 \int_{-\frac\pi2}^{\frac\pi2} \left( 4x\cos(x) + 1 + \cos(2x) \right) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%5Ciint_D%20%28x%20%2B%20y%29%20%5C%2C%20dA%20%3D%20%5Cfrac12%20%5Cint_%7B-%5Cfrac%5Cpi2%7D%5E%7B%5Cfrac%5Cpi2%7D%20%5Cleft%28%204x%5Ccos%28x%29%20%2B%201%20%2B%20%5Ccos%282x%29%20%5Cright%29%20%5C%2C%20dx)
Since 4x cos(x) is an odd function over the symmetric interval [-π/2, π/2], its contribution to the integral is 0, and the remaining integral is trivial.
![\displaystyle 2 \iint_D (x + y) \, dA = \frac12 \int_{-\frac\pi2}^{\frac\pi2} \left( 1 + \cos(2x) \right) \, dx = \boxed{\frac\pi2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%5Ciint_D%20%28x%20%2B%20y%29%20%5C%2C%20dA%20%3D%20%5Cfrac12%20%5Cint_%7B-%5Cfrac%5Cpi2%7D%5E%7B%5Cfrac%5Cpi2%7D%20%5Cleft%28%201%20%2B%20%5Ccos%282x%29%20%5Cright%29%20%5C%2C%20dx%20%3D%20%5Cboxed%7B%5Cfrac%5Cpi2%7D%20)
Is there a graph or a picture to go along with it? it would be easier for me to answer :)