Answer
$\dfrac{1}{2}+\ln (2)$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{1} (x-0) \ dx+\int_{1}^{2} (\dfrac{1}{x}-0) \ dx \\=[\dfrac{x^2}{2}]_0^{1}+[\ln (x)]_1^2 \\=[\dfrac{1}{2}-0]+[\ln 2-\ln 1]\\=\dfrac{1}{2}+\ln (2)$