The area (in sq. units) of the region

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 :

1. (1) $\frac{53}{6}$

2. (2) 8

3. (3) $\frac{59}{6}$

4. (4) $\frac{26}{3}$

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

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

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