The area of the region x,y:x^2≤y≤x^2-4,y≥1 is

Correct answer: (A) 4/3(4√2-1). The required diagram is: The shaded region is symmetric about y-axis. Required area =2∫12√ydy+∫24√(4-y)dy =2y^(3/2)3/212-(2(4-y)^(3/2))/324 =4/3(4√2-1) Hence this is the correct option.

Q42 / 90 · JEE Main 2023 (13 Apr Shift 2) · Mathematics

The area of the region $x, y : x^{2} \leq y \leq |x^{2} - 4|, y \geq 1$ is

Correct answer
A
$\frac{4}{3} (4 \sqrt{2} - 1)$
B
$\frac{4}{3} (4 \sqrt{2} + 1)$
C
$\frac{3}{4} (4 \sqrt{2} + 1)$
D
$\frac{3}{4} (4 \sqrt{2} - 1)$

Solution

The required diagram is:
The shaded region is symmetric about $y -$ axis.
Required area $= 2 [\int_{1}^{2} \sqrt{y} d y + \int_{2}^{4} \sqrt{4 - y} d y]$
$= 2 {[\frac{y^{3 / 2}}{\frac{3}{2}}]}_{1}^{2} - {[\frac{2 (4 - y)^{3 / 2}}{3}]}_{2}^{4}$
$= \frac{4}{3} (4 \sqrt{2} - 1)$
Hence this is the correct option.

Hence, the correct answer is option: 'A'.

The area of the region x,y:x2≤y≤x2-4,y≥1 is

Solution

Relevant Questions

The area of the region $x, y : x^{2} \leq y \leq |x^{2} - 4|, y \geq 1$ is