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Questions  >>  CBSE XII  >>  Math  >>  Application of Integrals

Find the area of the region bounded by $y=\sqrt x$ and $y=x.$

1 Answer

  • The area enclosed by the curve $y=f(x)$,the $x$-axis and the ordinates $x=a$ and $x=b$ is given by $\int_a^b ydx.$
  • $\int x^n dx=\large\frac{x^{n+1}}{n+1}$$+c$.
Step 1:
Given :$y=\sqrt x$ and $y=x$.
Consider $y=\sqrt x$
Squaring on both sides we get,
Clearly this is a parabola open rightward and whose focus is $(4a=1\Rightarrow a=\large\frac{1}{4}$)
$y=x$ is a straight line passing through the origin which is inclined at an angle of $45^\circ$ to the $x$-axis.
Step 2:
The points of intersection can be obtained by equating $y^2=x$ and $y^2=x^2$
$x^2=x\Rightarrow (x^2-x)=0$
$\Rightarrow x=0$ or $x=1$.Hence $y=0,1.
Hence the points of intersection are (0,0) and (1,1).
The required area is the shaded portion shown in the fig.
Hence $A=\int_0^1(y_2-y_1)dx.$
Where $y_2=\sqrt{x}$ and $y_1=x$
So,$A=\int_0^1(\sqrt x-x)dx.$
$\;\;\;\;=\int_0^1\sqrt{x}dx-\int_0^1 xdx.$
Step 3:
On integrating we get,
On applying limits we get,
Hence the required area is $\large\frac{1}{6}$sq.units.
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