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JEE Main 202224 JuneEvening Shift
Exam
JEE Main
Year
2022
Date
24 June
Shift
Evening Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2022 · 24 June · Evening Shift · Mathematics · Area Under Curves

The area (in sq. units) of the region enclosed between the parabola...

The area (in sq. units) of the region enclosed between the parabola y2=2x{y}^{2} = 2x and the line x+y=4x + y = 4 is ______.The area (in sq. units) of the region enclosed between the parabola y ^ 2 = 2x and the line x + y = 4 is ______.

Correct answer

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Step-by-step solution

The area (in sq. units) of the region enclosed between the parabola y ^ 2 = 2x and the line x + y = 4 is ______.

y2=2x⇒y2=2(4−y){y}^{2} = 2x ⇒ {y}^{2} = 2\left(4 - y\right)

⇒y2+2y−8=0⇒y=−4,2⇒ {y}^{2} + 2y - 8 = 0 ⇒ y = - 4, 2

The point of intersection of y2=2x{y}^{2} = 2x and x+y=4x + y = 4 are \left(8, - 4\right)and(2,2)and \left(2, 2\right)

Area =∫−42((4−y)−y22)dy=(4y−y22−y36)−42= {∫}_{- 4}^{2}\left(\left(4 - y\right) - \dfrac{{y}^{2}}{2}\right)dy = {\left(4y - \dfrac{{y}^{2}}{2} - \dfrac{{y}^{3}}{6}\right)}_{- 4}^{2}

=(8−2−43)−(−16−8+323)= \left(8 - 2 - \dfrac{4}{3}\right) - \left(- 16 - 8 + \dfrac{32}{3}\right)

=543=18= \dfrac{54}{3} = 18 Sq.units.

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