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JEE Main 2016 · 3 Apr · Shift 1, Mathematics question 76

The area (in sq. units) of the region {(x,y):y2≥2x and x2+y2≤4x, x≥0, y≥0}\{(x,y):y^2\ge2x\text{ and }x^2+y^2\le4x,\ x\ge0,\ y\ge0\} is :

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Answer and worked solution

Official answer: B

The circle (x−2)2+y2=4(x-2)^2+y^2=4 meets y2=2xy^2=2x at x=0x=0 and x=2x=2. The region lies above the parabola and inside the circle for 0≤x≤20\le x\le2:
∫024x−x2 dx−∫022x dx=π−83.\int_0^2\sqrt{4x-x^2}\,dx-\int_0^2\sqrt{2x}\,dx=\pi-\frac83.

Topics:Area between curves
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