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JEE Main 202313 AprilEvening Shift
Exam
JEE Main
Year
2023
Date
13 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2023 · 13 April · Evening Shift · Mathematics · Area Under Curves

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

The area of the region x,y:x2≤y≤∣x2−4∣,y≥1x, y: {x}^{2} \le y \le \left|{x}^{2} - 4\right|, y \ge 1 isThe area of the region x, y: x ^ 2 ≤ y ≤ | x ^ 2 - 4|, y ≥ 1 is

Options

  1. A43(42−1)\dfrac{4}{3} ( 4\sqrt{2} - 1 )
  2. B43(42+1)\dfrac{4}{3} ( 4\sqrt{2} + 1 )
  3. C34(42+1)\dfrac{3}{4} ( 4\sqrt{2} + 1 )
  4. D34(42−1)\dfrac{3}{4} ( 4\sqrt{2} - 1 )

Correct answer

A. 43(42−1)\dfrac{4}{3} ( 4\sqrt{2} - 1 )

Step-by-step solution

The required diagram is:

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

The shaded region is symmetric about y−y -axis.

Required area =2[∫12ydy+∫244−ydy]= 2\left[{∫}_{1}^{2}\sqrt{y}dy + {∫}_{2}^{4}\sqrt{4 - y}dy\right]

=2[y3/232]12−[2(4−y)3/23]24= 2{\left[\dfrac{{y}^{3 / 2}}{\dfrac{3}{2}}\right]}_{1}^{2} - {\left[\dfrac{2 ( 4 - y{)}^{3 / 2}}{3}\right]}_{2}^{4}

=43(42−1)= \dfrac{4}{3} ( 4\sqrt{2} - 1 )

Hence this is the correct option.

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