Tha answer would be C. Standardized tests will never ask you about anything not provided by them, so A, B, and D would not be feasible.
Gold and platinum are even known as noble metals. They are not affected by air, water and even by chemicals. Since they have bright lustre, jewellery can be made from them
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)
"Spanish conquistadors settled in the Caribbean in search of gold and riches to bring back for their country."
"Spanish conquistadors were explorer-soldiers who settled in the Caribbean with the aim to amass wealth and fortune. Christopher Columbus arrived in the Caribbean in 1492 and claimed the area for Spain. Spanish settlements began to sprout in the region in the following years. The Caribbean’s importance also included its strategic location.
From the early 17th century, non-Hispanic traders and settlers also arrived and established permanent colonies and trading posts in the various Caribbean islands, particularly in areas where Spanish power was weak or absent."
To learn more about the Caribbean: brainly.com/question/9781373
#SPJ4