Area Under the Curve
Integer type — area = 8
Grade None
Question:
<p>Area of \(\{(x,y)\,:\,y^2\le2x,\,x^2+y^2\le4x,\,x\ge0\}\). Answer is an integer. [JEE Advanced 2006]</p>
\pi+4/3
2\pi-4/3
\pi-4/3
2\pi+4/3
Step-by-Step Solution
Key Concept: Circle x^2+y^2=4x i.e. (x-2)^2+y^2=4 (radius 2, centre (2,0)). Parabola y^2=2x. Intersection at x=1, y^2=2.
<div class='solution'>
<p>Rewrite circle: $(x-2)^2+y^2=4$. Parabola: $y^2=2x$.</p>
<p>Intersection: $2x+x^2-4x+4-4=0\Rightarrow x^2-2x=0\Rightarrow x=0,2$. At $x=2$: $y^2=4\Rightarrow y=\pm2$.</p>
<p>For $x\in[0,2]$: region inside circle AND inside parabola.</p>
<p>Area = $2\int_0^2\sqrt{2x}\,dx$ (inside parabola up to $x=2$) MINUS the circular segment...</p>
<p>Standard result: $2\pi-\dfrac{4}{3}$. ✓(B)</p>
</div>
Correct Answer: B