Answer
$\dfrac{15}{8}-2 \log (2)$
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:
$A=\int_a^b [f(x)-g(x)] \ dx\\ = \int_{1/2}^{2} [\dfrac{5}{2}-\dfrac{1}{x}-x] \ dx \\= [\dfrac{5x}{2}-\log |x| -\dfrac{x^2}{2}]_{1/2}^{2} \\= [5 -\log 2 -2)-(\dfrac{5}{4}-\log \dfrac{1}{2}-\dfrac{1}{8}) \\=3-\log 2 -\dfrac{5}{4}+\log \dfrac{1}{2}+\dfrac{1}{8}\\=\dfrac{15}{8}-2 \log (2)$
