Let $R$ be the region bounded by the graph of $x = e ^ { y }$, the vertical line $x = 10$, and the horizontal lines $y = 1$ and $y = 2$. Which of the following gives the area of $R$?\\
(A) $\int _ { 1 } ^ { 2 } e ^ { y } \, d y$\\
(B) $\int _ { e } ^ { e ^ { 2 } } \ln x \, d x$\\
(C) $\int _ { 1 } ^ { 2 } \left( 10 - e ^ { y } \right) d y$\\
(D) $\int _ { e } ^ { 10 } ( \ln x - 1 ) d x$