The area (in sq. units) of the region bounded by the curve $x^2 = 250y, y = 0$ and $x = 50$ is

$\frac{500}{3}$

The correct answer is Option (4) → $\frac{500}{3}$

Given curves: $x^2 = 250y$, $y=0$, $x=50$

Rewrite $y$ in terms of $x$: $y = \frac{x^2}{250}$

Area $A = \int_{x=0}^{50} y \, dx = \int_0^{50} \frac{x^2}{250} \, dx = \frac{1}{250} \int_0^{50} x^2 \, dx$

$\int_0^{50} x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^{50} = \frac{125000}{3}$

So $A = \frac{1}{250} * \frac{125000}{3} = \frac{500}{3} \, \text{sq. units}$