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JEE Main 202127 JulyEvening Shift
Exam
JEE Main
Year
2021
Date
27 July
Shift
Evening Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2021 · 27 July · Evening Shift · Mathematics · Area Under Curves

The area of the region bounded by y - x = 2 and x ^ 2 = y is equal...

The area of the region bounded by y−x=2y - x = 2 and x2=y{x}^{2} = y is equal to :-The area of the region bounded by y - x = 2 and x ^ 2 = y is equal to :-

Options

  1. A163\dfrac{16}{3}
  2. B23\dfrac{2}{3}
  3. C92\dfrac{9}{2}
  4. D43\dfrac{4}{3}

Correct answer

C. 92\dfrac{9}{2}

Step-by-step solution

The area of the region bounded by y - x = 2 and x ^ 2 = y is equal to :-

y−x=2,x2=yy - x = 2, {x}^{2} = y

Now, x2=2+x{x}^{2} = 2 + x

⇒x2−x−2=0⇒ {x}^{2} - x - 2 = 0

⇒(x+1)(x−2)=0⇒x=−1,2⇒ \left(x + 1\right)\left(x - 2\right) = 0 ⇒ x = - 1, 2

Required area=∫−12(2+x−x2)dx= {∫}_{- 1}^{2}\left(2 + x - {x}^{2}\right)dx

=∣2x+x22−x33∣−12= {\left|2x + \dfrac{{x}^{2}}{2} - \dfrac{{x}^{3}}{3}\right|}_{- 1}^{2}

=(4+2−83)−(−2+12+13)= \left(4 + 2 - \dfrac{8}{3}\right) - \left(- 2 + \dfrac{1}{2} + \dfrac{1}{3}\right)

=6−3+2−12=92= 6 - 3 + 2 - \dfrac{1}{2} = \dfrac{9}{2} square units

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