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# Find the area of the region bounded by the parabola $y=x^2$ and y=| x |.

Given $y=x^2\; and\; y=x^2$
$x^2=y$ represents a parabola with vertex (0,0),open upwards, in the positive direction.
$y=x,$ represents a line passing through the origin making an angle $45^{\circ}$ in the positive direction of x-axis and
$y=-x,$ represents a line passing through the origin making an angle $135^{\circ}$ in the positive direction of x-axis
$A=2\int \limits_0^1 (x-x^2)dx$
$=2 \int \limits_0^1 x-2 \int \limits_0^1 x^2 dx$
$=2 \bigg[\frac{x^2}{2}-\frac{x^3}{3} \bigg]_0^1$
$=2\bigg[\frac{1}{2}-\frac{1}{3}\bigg]=2 \times \frac{1}{6}=\frac{1}{3} sq.units$
edited Apr 3, 2013 by meena.p