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

Explanation

Given parabola `y=x^(2)` is symmetric about Y-axis and passes through the point (0, 0) .

` :. ` The curve y = |x| is symmetric about Y-axis and passes through the origin (0,0).
The region bounded by the curve `y=x^(2)` and lines y = |x| i.e., `y=+- x ` is shown in the figure . The point of intersection of the parabola `x^(2) =y` and `y=x` in first quadrant is point A(1, 1). Given region is bounded about Y-axis.
` :.` ar(OACO)`=` ar(ODBO)
` :. ` Required area `=2 xx`(area of shaded region in first quadrant only)
`=2 int_(0)^(1)(y_(2)-y_(1))dx=2 int_(0)^(1)(x-x^(2))dx`
`=2[int_(0)^(1)xdx-int_(0)^(1)x^(2)dx]`
`=2([(x^(2))/(2)]_(0)^(1)-[(x^(3))/(3)]_(0)^(1))`
`=2{((1)/(2)-0)-((1)/(3)-0)}=(1)/(3)` sq. unit.
Thereforre, required area is `(1)/(3)` sq. unit.