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What is the integral that represents the surface area of the surface with parameterization $\vec{G}(x.y) = ( x, y, e^{-x}\cos y )$ for $0 \leq x \leq 1$ and $0 \leq y \leq 2\pi$.

A

$\int_0^{2\pi} \int_0^1 \sqrt{1+e^{-2x}} \, dx \, dy$

B

$\int_0^1 \int_0^{2\pi} \sqrt{1+e^{-2x}} \, dx \, dy$

C

$\int_0^{2\pi} \int_0^1 (1+e^{-x}) \, dx \, dy$

D

$\int_0^{2\pi} \int_0^1 e^{-x} \, dx \, dy$

E

$\int_0^{2\pi} \int_0^1 \left(1+e^{-x}(\sin y + \cos y)\right) \, dx \, dy$

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