Answer
$\dfrac{125}{6}$
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:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{-1}^{4} [3y-(y^2-4)] \ dy \\=[\dfrac{3y^2}{2}-\dfrac{y^3}{3}+4y]_{-1}^4 \\=[24-\dfrac{64}{3}+16]-[\dfrac{3}{2}+\dfrac{1}{3}-4]\\=\dfrac{112+13}{3}\\=\dfrac{125}{6}$