Question

The area of the region bounded by the parabola $y=x^2$ and $y$ $=|x|$ is

• A. $3$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $2$

Answer 1

Answer: (C)

Clearly $x^2=y$ represents a parabola with vertex at $(0,0)$ positive direction of y-axis as its axis opens upwards.
$y=|x|$ i.e., $y=x$ and $y=-x$ represent two lines passing through the origin and making an angle of $45^{\circ }$ and $135^{\circ }$ with the positive direction of the $x$ -axis.

The required region is the shaded region as shown in the figure. Since both the curve are symmetrical about y-axis. So, required area $=2$ (shaded area in the first quardant)
$-2\underset{0}{\overset{1}{\int }}\left(x-x^2\right)dx=2{\left[\frac{x^2}{2}-\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$ sq. units