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Let $R$ be the region in the first quadrant bounded by the graphs of $y=e^{x}$, $x=1$ and the $x$ and $y$ axes. If $R$ is revolved about the horizontal line $y$ = $e$, the volume of the resulting solid is represented by which of the following integral expressions?

A

$\pi \int_{0}^{1}e^{2}-\left ( e-e^{x} \right )^{2}dx$

B

$\pi\int_{0}^{1}e^{2x}dx$

C

$\pi\int_{0}^{1}\left ( e-e^{x} \right )^{2}dx$

D

$\pi\int_{0}^{1} \ e^{2}-e^{2x}dx$

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