Question

Area of the region enclosed by the curves $y=x^2$ and $y=x^3$ is

• A. $\frac{1}{12}$ sq unit
• B. $\frac{1}{6}$ sq unit
• C. $\frac{1}{3}$ sq unit
• D. 1 sq unit

Answer 1

Answer: (A)

We have,
Equation of the given curves as
$y=x^2......$(i)
$y=x^3......$(ii)
On equating Eqs. (i) and (ii), we get
$x^3=x^2$
$\Rightarrow x^3-x^2=0$
$\Rightarrow x^2(x-1)=0$
$\Rightarrow x=0$ or $1$

Intersection points of given curves are $(1,1)$ and $(0,0)$.
Now, sketch the graph of the given curves and shade the common region.
Area of shaded region $=$ Area $(OCBAO)-$ Area $(OBAO)$
$\therefore \text{ Area }=\underset{0}{\overset{1}{\int }}\left(x^2-x^3\right)dx$
$={\left[\frac{x^3}{3}-\frac{x^4}{4}\right]}_0^1=\frac{1}{12}\text{ sq unit }$