# Q : 9         Find the area of the region bounded by the parabola $\dpi{100} \small y=x^2$ and $\dpi{100} \small y=|x|$.

G Gautam harsolia

We can clearly see that given area is symmetrical about y-axis
Therefore,
Area of OCAO = Area of OBDO
Point of intersection of $y=x^2 \ and \ y = |x|$  is  (1 , 1)  and (-1 , 1)
Now,
Area od OCAO = Area OAM - Area of OCMO
Area of OAM = $\frac{1}{2}.OM.AM = \frac{1}{2}.1.1 = \frac{1}{2}$
Area of OCMO = $\int_{0}^{1}ydx= \int_{0}^{1}x^2dx= \left [ \frac{x^3}{3} \right ]_{0}^{1}= \frac{1}{3}$
Therefore,
Area od OCAO $=\frac{1}{2}- \frac{1}{3}= \frac{1}{6}$
Now,
Area of the region bounded by the parabola $\dpi{100} \small y=x^2$ and $\dpi{100} \small y=|x|$ is = 2 X Area od OCAO  $=2\times \frac{1}{6} = \frac{1}{3}$  Units

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