Answer
$2+\dfrac{\pi^3}{6}$
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:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{\pi} [\sin x-x(x-\pi)] \ dx \\=(-\cos x-\dfrac{x^3}{3}+\dfrac{x^2\pi}{2}]_{0}^{\pi} \\=[-(-1)-\dfrac{\pi^3}{3}+\dfrac{\pi^3}{2}]-(-1) \\=2-\pi^3(\dfrac{1}{3}-\dfrac{1}{2}) \\=2+\dfrac{\pi^3}{6}$
