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Q.
The area of the region bounded by the parabola $y=x^{2}$ and $y$ $=| x |$ is
Application of Integrals
Solution:
Clearly $x ^{2}= y$ represents a parabola with vertex at $(0,0)$ positive direction of y-axis as its axis opens upwards.
$y =| x |$ i.e., $y = x$ and $y =- x$ represent two lines passing through the origin and making an angle of $45^{\circ}$ and $135^{\circ}$ with the positive direction of the $x$ -axis.
The required region is the shaded region as shown in the figure. Since both the curve are symmetrical about y-axis. So, required area $=2$ (shaded area in the first quardant)
$-2 \int\limits_{0}^{1}\left(x-x^{2}\right) d x=2\left[\frac{x^{2}}{2}-\frac{x^{3}}{3}\right]_{0}^{1}=\frac{1}{3}$ sq. units
