Answer
$$A=2$$
Work Step by Step
To find the area of the region, we consider the following integral.
\begin{align}
A&=\int_{x=2}^{x=5} \int_{y=0}^{y=\frac{1}{ \sqrt{x-1}}} d y \ d x\\
&=\int_{x=2}^{x=5}[y]_{0}^{\frac{1}{ \sqrt{x-1}}} d x\\
&=\int_{x=2}^{x=5} \frac{1}{\sqrt{x-1}} d x\\
&=\int_{x=2}^{x=5} (x-1)^{-\frac{1}{2}} \ d x\\
&= [2 (x-1)^{-\frac{1}{2}+1}]_{2}^{5} \\
&=[2 \sqrt{x-1}]_{2}^{5}\\
&=2 \sqrt{4}-2 \sqrt{1}\\
&=2
\end{align}
