Question

Find the area between the curve $y=x^2$ and $y=x$.

• (A) $\frac{5}{6}$ sq. unit
• (B) $\frac{1}{3}$ sq. unit
• (C) $\frac{1}{2}$ sq. unit
• (D) $\frac{1}{6}$ sq. unit

Answer 1

The Correct Option is D

Explanation:
Given:Curve $1:y=x^2=f(x)$ (say)Curve 2 : $\mathrm{y}-\mathrm{x}=0$$\Rightarrow y=x=g(x)$ (say)The curve can be drawn as:

To find the intersections (or limits of the area) putting value of $y$ from curve 1$\Rightarrow {\mathrm{x}}^2-\mathrm{x}=0$$\Rightarrow \mathrm{x}(\mathrm{x}-1)=0$$\Rightarrow \mathrm{x}=0,\mathrm{x}=1$Now the required area $(A)$ is:$A=|{\int }_{x_1}^{x_2}[f(x)-g(x)]dx|$$\Rightarrow A=\left|{\int }_0^1[x^2-x]dx\right|$$\Rightarrow A=\left|{[\frac{x^3}{3}-\frac{x^2}{2}]}_0^1\right|$$\Rightarrow A=|\frac{1}{3}-\frac{1}{2}-(0)|$$\Rightarrow A=\frac{1}{6}$ sq. unitHence, the correct option is (D).