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JEE Main 202125 JulyMorning Shift
Exam
JEE Main
Year
2021
Date
25 July
Shift
Morning Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2021 · 25 July · Morning Shift · Mathematics · Area Under Curves

The area (in sq. units) of the region, given by the set \ (x, y) ∈...

The area (in sq. units) of the region, given by the set {(x,y)∈R×R∣x≥0,2x2≤y≤4−2x}\left\{\left(x, y\right) \in R \times R ∣ x \ge 0, 2{x}^{2} \le y \le 4 - 2x\right\} is :The area (in sq. units) of the region, given by the set \ (x, y) ∈ R × R ∣ x ≥ 0, 2 x ^ 2 ≤ y ≤ 4 - 2x\ is :

Options

  1. A83\dfrac{8}{3}
  2. B173\dfrac{17}{3}
  3. C133\dfrac{13}{3}
  4. D73\dfrac{7}{3}

Correct answer

D. 73\dfrac{7}{3}

Step-by-step solution

The area (in sq. units) of the region, given by the set \ (x, y) ∈ R × R ∣ x ≥ 0, 2 x ^ 2 ≤ y ≤ 4 - 2x\ is :

Given curves are y=2x2 and y=4−2xy = 2{x}^{2} \text{ and } y = 4 - 2x

So, 2x2=4−2x2{x}^{2} = 4 - 2x

⇒2x2+2x−4=0⇒(x−1)(x+2)=0⇒ 2{x}^{2} + 2x - 4 = 0 ⇒ \left(x - 1\right)\left(x + 2\right) = 0

So, the xx-coordinate of the intersection point is 11 as x=−2<0x = - 2 < 0

Required area ∫01(4−2x−2x2)dx=[4x−x2−2x33]01{∫}_{0}^{1}\left(4 - 2x - 2{x}^{2}\right)dx = {\left[4x - {x}^{2} - \dfrac{2{x}^{3}}{3}\right]}_{0}^{1}

=4−1−23=73= 4 - 1 - \dfrac{2}{3} = \dfrac{7}{3}

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