Question

The area of the region bounded by $y=x$ and $y=x^2$ is ______.

• A. $\frac{2}{3}$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\frac{1}{6}$

Answer 1

Answer: (D)

$\frac{1}{6}$

First we find the cordinates of point P

$y=x$ and $y=x^2$

$⟹x=x^2$

$⟹x=0,1$

The value $x=0$ correspond to $O(0,0)$.

For $P$, $x=1$

Now, the required area

$A={\int }_0^1(y_1-y_2)dx$

$={\int }_0^1(x-x^2)dx$

$={[\frac{x^2}{2}-\frac{x^3}{3}]}_0^1$

$⟹A=\frac{1}{6}$