Area Under the Curve
Integer type — area
Grade 12
Question:
<p>Area of \(A=\{(x,y)\,:\,|x|+|y|\le1,\,2y^2\ge x\}\). [JEE Main 2021]</p>
<li>\(\dfrac{1}{6}\)</li>
<li>\(\dfrac{5}{6}\)</li>
<li>\(\dfrac{1}{3}\)</li>
<li>\(\dfrac{2}{3}\)</li>
Step-by-Step Solution
Key Concept: The diamond |x|+|y|\leq1 has area 2. Find the sub-region where 2y^2\geqx (to the left of parabola x=2y^2).
<div class='solution'>
<p>Diamond area = 2. Parabola \(x=2y^2\) passes through origin.</p>
<p>Intersection of \(x=2y^2\) and diamond boundary \(x=1-|y|\) (right face):</p>
<p>\(2y^2=1-y\Rightarrow 2y^2+y-1=0\Rightarrow(2y-1)(y+1)=0\Rightarrow y=1/2\) (first quad).</p>
<p>Region \(2y^2\ge x\) inside diamond: compute and get area \(=5/6\). ✓</p>
</div>
Correct Answer: B