Question

Find the area bounded by curves $\left\{\right(x,y):y\ge x^{2}andy\le |x|$

Answer 1

Given the curve $y=x^2$ ...(i)
and equation of line y = |x|
In first quadrant, the equation of the line is y = x and in second quadrant, the equation of line is y = - x.
The points of intersection of the parabola and line are (-1, 1) and (1, 1). As the area to be found is symmetrical about Y-axis.
$\therefore$ Required area = 2 (Area of shaded region in the first quadrant)
$=2{\int }_0^1(x-x^2)dx=2[{\int }_0^{1}xdx-{\int }_0^1{x}^{2}dx]=2\left(\right[\frac{x^2}{2}{]}_0^1-\left[\frac{x^3}{3}{]}_0^1\right)=2\left[\right(\frac{1}{2}-0)-(\frac{1}{3}-0\left)\right]=2[\frac{1}{2}-\frac{1}{3}]=2\left[\frac{1}{6}\right]=\frac{1}{3}squnit$