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JEE Main 20206 SeptemberMorning Shift
Exam
JEE Main
Year
2020
Date
6 September
Shift
Morning Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2020 · 6 September · Morning Shift · Mathematics · Area Under Curves

The area (in sq. units) of the region A = \ ( x, y ) : | x | + | y...

The area (in sq. units) of the region A={(x,y):∣x∣+∣y∣≤1,2y2≥∣x∣}A = \left\{( x, y ) : | x | + | y | \le 1, 2{y}^{2} \ge | x |\right\}
The area (in sq. units) of the region A = \ ( x, y ) : | x | + | y | ≤ 1, 2 y ^ 2 ≥ | x |\

Options

  1. A13\dfrac{1}{3}
  2. B76\dfrac{7}{6}
  3. C16\dfrac{1}{6}
  4. D56\dfrac{5}{6}

Correct answer

D. 56\dfrac{5}{6}

Step-by-step solution

The area (in sq. units) of the region A = \ ( x, y ) : | x | + | y | ≤ 1, 2 y ^ 2 ≥ | x |\

Required area is, A=ar(OAC)+ar(δABC)A = ar \left(OAC\right) + ar \left(\delta ABC\right)

⇒A=4[∫01/22y2dy+12⋅(12)⋅(12)]⇒ A = 4\left[{∫}_{0}^{1 / 2}2{y}^{2}dy + \dfrac{1}{2} \cdot \left(\dfrac{1}{2}\right) \cdot \left(\dfrac{1}{2}\right)\right]

⇒A=4[(23y3)01/2+(18)]=56⇒ A = 4\left[{\left(\dfrac{2}{3}{y}^{3}\right)}_{0}^{1 / 2} + \left(\dfrac{1}{8}\right)\right] = \dfrac{5}{6}

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