Answer
$\dfrac{5}{24}$
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:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{\sqrt 3/2} (2x\sqrt {1-x^2}-x)\ dx \\=[-\dfrac{2(1-x^2)^{3/2}}{3}-\dfrac{x^2}{2}]_{0}^{\sqrt 3/2} \\=[-\dfrac{2(1-\dfrac{3}{4})^{3/2}}{3}-\dfrac{3}{8}]-[-\dfrac{2(1)^{3/2}}{3}-0]\\=\dfrac{-2}{24}-\dfrac{3}{8}+\dfrac{2}{3}\\=\dfrac{5}{24}$
