The area of the region bounded by the curves $y = x^2 + 2$ and x-axis, between $x = 0$ and $x = 3$ in the first quadrant is:

15 sq.units

The correct answer is Option (2) → 15 sq.units

Given curves: $y = x^2 + 2$ and x-axis, between $x=0$ and $x=3$

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

$\int (x^2 + 2) dx = \int x^2 dx + \int 2 dx = \frac{x^3}{3} + 2x$

Evaluate from 0 to 3:

$\left[ \frac{3^3}{3} + 2*3 \right] - \left[ \frac{0^3}{3} + 2*0 \right] = (9 + 6) - 0 = 15$

Area of the region = 15 square units