The area of the region enclosed by $\mathrm{y} \leq 4 \mathrm{x}^{2}, \mathrm{x}^{2} \leq 9 \mathrm{y}$ and $\mathrm{y} \leq 4$, is equal to :

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Area Under Curves

The area of the region enclosed by

y \leq 4 x^{2}, x^{2} \leq 9 y

and

y \leq 4

, is equal to :

Answer: (D)

The region is bounded by the parabola

y = 4 x^{2}

(opening upward),

x^{2} = 9 y

(i.e.,

y = \frac{x ^{2}}{9}

, opening upward but wider), and the horizontal line

y = 4

.

From

x^{2} = 9 y

, we get

x = 3 y

. From

y = 4 x^{2}

, we get

x = \frac{y}{2}

.

The area (using symmetry about the y-axis) is:

Δ = 2 \int_{0}^{4} (3 y - \frac{y}{2}) d y = 2 \int_{0}^{4} \frac{5 y}{2} d y = 5 \int_{0}^{4} y d y

= 5 [\frac{2 y ^{3/2}}{3}]_{0}^{4} = 5 \cdot \frac{2 \cdot 8}{3} = \frac{80}{3}

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Single correct option