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JEE Main 202324 JanuaryMorning Shift
Exam
JEE Main
Year
2023
Date
24 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2023 · 24 January · Morning Shift · Mathematics · Area Under Curves

The area enclosed between the curves y ^ 2 + 4x = 4 and y - 2x = 2...

The area enclosed between the curves y2+4x=4{y}^{2} + 4x = 4 and y−2x=2y - 2x = 2 isThe area enclosed between the curves y ^ 2 + 4x = 4 and y - 2x = 2 is

Options

  1. A253\dfrac{25}{3}
  2. B223\dfrac{22}{3}
  3. C99
  4. D233\dfrac{23}{3}

Correct answer

C. 99

Step-by-step solution

Given:

y2=−4x+4{y}^{2} = - 4x + 4

y=2x+2y = 2x + 2

The area enclosed between the curves y ^ 2 + 4x = 4 and y - 2x = 2 is

Required area

=∫−42(4−y24−y−22)dy= {∫}_{- 4}^{2}\left(\dfrac{4 - {y}^{2}}{4} - \dfrac{y - 2}{2}\right)dy

=∫−42(2−y24−y2)dy= {∫}_{- 4}^{2}\left(2 - \dfrac{{y}^{2}}{4} - \dfrac{y}{2}\right)dy

=[2y−y312−y24]−42= {\left[2y - \dfrac{{y}^{3}}{12} - \dfrac{{y}^{2}}{4}\right]}_{- 4}^{2}

=[(4−812−1)−(−8+6412−164)]= \left[\left(4 - \dfrac{8}{12} - 1\right) - \left(- 8 + \dfrac{64}{12} - \dfrac{16}{4}\right)\right]

=15−6= 15 - 6

=9= 9 sq. units

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