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Q.
Area of the region enclosed by the curves $y=x^2$ and $y=x^3$ is
Application of Integrals
Solution:
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 $(O C B A O)-$ Area $(O B A O)$
$\therefore \text { Area } =\int\limits_0^1\left(x^2-x^3\right) d x $
$ =\left[\frac{x^3}{3}-\frac{x^4}{4}\right]_0^1=\frac{1}{12} \text { sq unit }$
