Find the area of the shaded region between the curve, $y = 4 - x^2$, $0 \leq x \leq 3$ and the $X$-axis.

$\frac{23}{3}$

The correct answer is Option (3) → $\frac{23}{3}$

Required area is given by the absolute value of the integral $I$, where:

$I = \int_{0}^{2} (4 - x^2) \, dx + \int_{2}^{3} -(4 - x^2) \, dx$

$= \left[ 4x - \frac{x^3}{3} \right]_{0}^{2} - \left[ 4x - \frac{x^3}{3} \right]_{2}^{3}$

$= \left( 8 - \frac{8}{3} \right) - 0 - \left[ (12 - 9) - \left( 8 - \frac{8}{3} \right) \right]$

$= \frac{16}{3} - \left[ 3 - \frac{16}{3} \right]$

$= \frac{23}{3} \text{ sq. units}$