First slide

Area of bounded Regions

Question

The area of the region, enclosed by the circle $x^{2} + y^{2} = 2$ which is not common to the region bounded by the parabola $y^{2} = x$ and the straight line y = x, is:

Moderate

A

$\frac{1}{6} (24 π - 1)$
B

$\frac{1}{6} (12 π - 1)$
C

$\frac{1}{3} (6 π - 1)$
D

$\frac{1}{3} (12 π - 1)$

Solution

The requireed area is the area of the region bounded by the circle minus the area of the region bounded by $y^{2} = x$ and the line $y = x$
Required area =Total area – enclosed area =   $2 π - \int_{0}^{1} \sqrt{x} - x d x$
   $= 2 π - {(\frac{2 x^{3 / 2}}{3} - \frac{x^{2}}{2})}_{0}^{1}$
   $= 2 π - (\frac{2}{3} - \frac{1}{2}) = 2 π - (\frac{1}{6}) = \frac{12 π - 1}{6}$

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