Question

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

Answer 1

The area bounded by the parabola, $x^2=y$, and the line, $y=|x|$, can be represented as
The given area is symmetrical about $y$-axis
$\therefore$ Area $OACO$ $=$ Area $ODBO$
The point of intersection of parabola, $x^2=y$, and line, $y=x$, is $A(1,1)$.
Area of $OACO$ $=$ Area $\Delta OAB-\text{Area}OBACO$
$\therefore \text{Area of}\Delta OAB=\frac{1}{2}\times OB\times AB=\frac{1}{2}\times 1\times 1=\frac{1}{2}$
Area of $OBACO$ $={\int }_0^{1}ydx={\int }_0^1{x}^{2}dx={\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$
$\Rightarrow$ Area of $OACO$ $=$ Area of $\Delta OAB-$ Area of $OBACO$
$=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$
Therefore, required area $=2\left[\frac{1}{6}\right]=\frac{1}{3}$ sq.units.