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JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Area Under Curves

The area (in sq. units) of the region described by \ (x, y): y^2 ≤...

The area (in sq. units) of the region described by {(x,y):y2≤2x, and y≥4x−1}\left\{(x, y): y^2 \leq 2 x\right.,\text{ and } \left.y \geq 4 x-1\right\} is

Options

  1. A1132\frac{11}{32}
  2. B89\frac{8}{9}
  3. C1112\frac{11}{12}
  4. D932\frac{9}{32}

Correct answer

D. 932\frac{9}{32}

Step-by-step solution

 Shaded area =∫−121(xRight −xLeft )dyy2=2xy=4x−1 Solve y=1,y=−12 Shaded area =∫−121(y+14−y22)dy=(14(y22+y)−y36)−121=932\begin{aligned} & \text{ Shaded area }=\int_{-\frac{1}{2}}^1\left(x_{\text{Right }}-x_{\text{Left }}\right) d y \\ & y^2=2 x \\ & y=4 x-1 \quad \text{ Solve } \\ & y=1, y=-\frac{1}{2} \\ & \text{ Shaded area }=\int_{-\frac{1}{2}}^1\left(\frac{y+1}{4}-\frac{y^2}{2}\right) d y \\ & =\left(\frac{1}{4}\left(\frac{y^2}{2}+y\right)-\frac{y^3}{6}\right)_{-\frac{1}{2}}^1=\frac{9}{32}\end{aligned}
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