Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202228 JuneMorning Shift
Exam
JEE Main
Year
2022
Date
28 June
Shift
Morning Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2022 · 28 June · Morning Shift · Mathematics · Area Under Curves

The area of the region S = \ (x, y): y ^ 2 ≤ 8x, y ≥ 2 x, x ≥ 1\ is

The area of the region S={(x,y):y2≤8x,y≥2x,x≥1}S = \left\{\left(x, y\right): {y}^{2} \le 8x, y \ge \sqrt{2}x, x \ge 1\right\} isThe area of the region S = \ (x, y): y ^ 2 ≤ 8x, y ≥ 2 x, x ≥ 1\ is

Options

  1. A526\dfrac{5\sqrt{2}}{6}
  2. B1926\dfrac{19\sqrt{2}}{6}
  3. C1326\dfrac{13\sqrt{2}}{6}
  4. D1126\dfrac{11\sqrt{2}}{6}

Correct answer

D. 1126\dfrac{11\sqrt{2}}{6}

Step-by-step solution

Plotting the region for the given inequalities y2≤8x,y≥2x,x≤1{y}^{2} \le 8x, y \ge \sqrt{2}x, x \le 1, we get

The area of the region S = \ (x, y): y ^ 2 ≤ 8x, y ≥ 2 x, x ≥ 1\ is

Required area will be ∫14(22x−2x)dx{∫}_{1}^{4}\left(2\sqrt{2}\sqrt{x} - \sqrt{2}x\right)dx

=22(x323/2)14−2(x22)14= 2\sqrt{2}{\left(\dfrac{x\dfrac{3}{2}}{3 / 2}\right)}_{1}^{4} - \sqrt{2}{\left(\dfrac{{x}^{2}}{2}\right)}_{1}^{4}

=423(8−1)−23(16−1)= \dfrac{4\sqrt{2}}{3}\left(8 - 1\right) - \dfrac{\sqrt{2}}{3}\left(16 - 1\right)

=2823−1522=1126= \dfrac{28\sqrt{2}}{3} - \dfrac{15\sqrt{2}}{2} = \dfrac{11\sqrt{2}}{6}

Practice Area Under Curves on Quantrex Academy →

More from Area Under Curves

Passage: Consider the curve C_1 given by y = e^ -x for x ∈ [0, 10π] , and the curve C_2… JEE Advanced 2026Passage: Consider the ellipses given by x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . Question: If… JEE Advanced 2026The area of the region \ (x, y) : x^2 - 8x ≤ y ≤ -x\ is : JEE Main 2026The area of the region \ (x, y) : 0 ≤ y ≤ 6 - x, y^2 ≥ 4x - 3, x ≥ 0\ is: JEE Main 2026The area of the region R = \ (x, y): xy ≤ 27, 1 ≤ y ≤ x^2\ is equal to: JEE Main 2026The area of the region bounded by the curves x+3y^2=0 and x+4y^2=1 is equal to: JEE Main 2026The area of the region \ (x, y): y ≤ π - |x|, y ≤ |x x|, y ≥ 0\ is: JEE Main 2026If the area of the region bounded by 16x^2 - 9y^2 = 144 and 8x - 3y = 24 is A, then 3(A +… JEE Main 2026Let P_ 1 : y=4 x^ 2 and P_ 2 : y=x^ 2 +27 be two parabolas. If the area of the bounded… JEE Main 2026The area of the region R=\ (x, y): x y ≤ 8,1 ≤ y ≤ x^ 2 , x ≥ 0\ is JEE Main 2026Let f(α) denote the area of the region in the first quadrant bounded by x=0, x=1, y^ 2 =x… JEE Main 2026Let A_ 1 be the bounded area enclosed by the curves y=x^ 2 +2, x+y=8 and y -axis that… JEE Main 2026The area of the region enclosed between the circles x^ 2 +y^ 2 =4 and x^ 2 +(y-2)^ 2 =4… JEE Main 2026Let the area of the region bounded by the curve y= \ x, x\ , lines x=0, x=(3 π)/(2) , and… JEE Main 2026The area of the region A=\ (x, y): 4 x^ 2 +y^ 2 8. and .y^ 2 4 x\ is: JEE Main 2026Let the line x=-1 divide the area of the region \ (x, y): 1+x^ 2 ≤ y ≤ 3-x\ in the ratio… JEE Main 2026 Full Area Under Curves list Area Under Curves JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs