Question

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

Answer 1

Given parabola $y=x^2$ which symmetrical about Y-axis and passes through (0, 0) and the curve y = |x|.

On putting x = - x, we get y = |-x| = |x|

$\therefore$ Curve y = |x| is symmetrical about Y-axis and passes through origin..

The area bounded by the parabola, $y=x^2$ and the line y = |x| or $y=\pm x$ can be represented in the figure.

The point of intersection of parabola, $x^2=y$ and line, y = x in first quadrant is A (1, 1). The given area is symmetrical about Y-axis. $\therefore$ Area OACO = Area ODBO

Required area = 2 (Area of shaded region in the first quadrant only)

$=2{\int }_0^1(y_2-y_1)dx=2{\int }_0^1(x-x^2)dx$

$(\because$ The curve y = |x| lies above the curve $y=x^2$ in [0, 1] so, we take $y_2=x$ and $y_1=x^2)$

$=2[{\int }_0^{1}xdx-{\int }_0^1{x}^{2}dx]=2\left(\right[\frac{x^2}{2}{]}_0^1-\left[\frac{x^3}{3}{]}_0^1\right)=2\left\{\right(\frac{1}{2}-0)-(\frac{1}{3}-0\left)\right\}=\frac{1}{3}squnit$
Therefore, required area is $\frac{1}{3}$ sq unit.