Let $a \in \mathbb R$ where $y = e^x$ at $(a, e^a)$ and $(a+1, e^{a+1})$. Use a double integral to find the area enclosed

Question

Let $a \in \mathbb R$ where $y = e^x$ at $(a, e^a)$ and $(a+1, e^{a+1})$. Use a double integral to find the area enclosed

would the area be given by

$$\int_{e^{a}}^{e^{a+1}} \int_{a+1}^{a} e^{x}dxdy$$?

Answer 1

The elemental area is $dx dy$. For every value of $x$, $y$ ranges from $0\rightarrow e^x$. Finally, $x$ ranges from $a\rightarrow a+1$. The total area is $\int_{a}^{a+1}\int_{0}^{e^x} dy dx$.