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

The region is bounded by the curve $x^2 = 250y$, the x-axis $y = 0$, and the vertical line $x = 50$.

The curve $x^2 = 250y$ can be rewritten as:

$y = \frac{x^2}{250}$

The parabola passes through the origin, so the region extends from $x = 0$ to $x = 50$.

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The area is given by:

$\text{Area} = \int_0^{50} y \, dx$

$= \int_0^{50} \frac{x^2}{250} \, dx$

$= \frac{1}{250} \int_0^{50} x^2 \, dx$

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Using the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1}$:

$= \frac{1}{250} \left[\frac{x^3}{3}\right]_0^{50}$

$= \frac{1}{250} \left(\frac{50^3}{3} - \frac{0^3}{3}\right)$

$= \frac{1}{250} \times \frac{50^3}{3}$

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Calculate $50^3 = 125,000$:

$= \frac{1}{250} \times \frac{125,000}{3}$

$= \frac{125,000}{750}$

$= \frac{500}{3}$

Therefore, the area of the region is $\frac{500}{3}$ square units.