The area (in sq. units) of the region

Since, the relation $y \leq x^{2}+3 x$ represents the region below the parabola in the $1^{\text {st }}$ quadrant

$\because y=4$

$\Rightarrow x^{2}+3 x=4 \Rightarrow x=1,-4$

$\therefore$ the required area $=$ area of shaded region

$=\int_{0}^{1}\left(x^{2}+3 x\right) d x+\int_{1}^{3} 4 \cdot d x=\left[\frac{x^{3}}{3}+\frac{3 x^{2}}{2}\right]_{0}^{1}+[4 x]_{1}^{3}$

$=\frac{1}{3}+\frac{3}{2}+8=\frac{59}{6}$