Answer
$\dfrac{111}{12}$
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:
$A=\int_a^b [f(x)-g(x)] \ dx\\ = \int_{-1}^{0} [\dfrac{7x+10}{3}+x^3] \ dx+ \int_0^2 [\dfrac{7x+10}{3}-x^3] \ dx \\= [\dfrac{7x^2}{3}+\dfrac{10x}{3}+\dfrac{x^4}{4}]_{-1}^0 + [\dfrac{7x^2}{3}+\dfrac{10x}{3}-\dfrac{x^4}{4}]_{0}^2\\= [0-(\dfrac{7}{6}-\dfrac{10}{3}+\dfrac{1}{4}]+ [\dfrac{14}{3}+\dfrac{20}{3}-4-0]\\=\dfrac{111}{12}$
