Question:
The area (in sq. units) of the region $\mathrm{A}=\left\{(x, y) \in \mathrm{R} \times \mathrm{R} \mid 0 \leq x \leq 3,0 \leq y \leq 4, y \leq x^{2}+3 x\right\}$ is :
Correct Option: , 3
Solution:
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
$=\int_{0}^{1}\left(x^{2}+3 x\right) d x+\int_{1}^{3} 4 \cdot d x=\left[\frac{x^{3}}{3}+\frac{3 x^{2}}{2}\right]_{0}^{1}+[4 x]_{1}^{3}$
$=\frac{1}{3}+\frac{3}{2}+8=\frac{59}{6}$