CUET UG 2025 — Mathematics Calculus

The region is bounded by the curve $x^2 = 250y$x2=250y, the x-axis $y = 0$y=0, and the vertical line $x = 50$x=50.

The curve $x^2 = 250y$x2=250y can be rewritten as:

$$y = \frac{x^2}{250}$$y=250x2​

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

---

The area is given by:

$$\text{Area} = \int_0^{50} y \, dx$$Area=∫050​ydx

$$= \int_0^{50} \frac{x^2}{250} \, dx$$=∫050​250x2​dx

$$= \frac{1}{250} \int_0^{50} x^2 \, dx$$=2501​∫050​x2dx

---

Using the power rule $\displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1}$∫xndx=n+1xn+1​:

$$= \frac{1}{250} \left[\frac{x^3}{3}\right]_0^{50}$$=2501​[3x3​]050​

$$= \frac{1}{250} \left(\frac{50^3}{3} - \frac{0^3}{3}\right)$$=2501​(3503​−303​)

$$= \frac{1}{250} \times \frac{50^3}{3}$$=2501​×3503​

---

Calculate $50^3 = 125,000$503=125,000:

$$= \frac{1}{250} \times \frac{125,000}{3}$$=2501​×3125,000​

$$= \frac{125,000}{750}$$=750125,000​

$$= \frac{500}{3}$$=3500​

Therefore, the area of the region is $\frac{500}{3}$3500​ square units.