Question

Area bounded by the curves $y=x^2$ and $y=2-x^2$ is

• A. $\frac{8}{3}$ sq units
• B. $\frac{3}{8}$ sq units
• C. $\frac{3}{2}$ sq units
• D. None of these

Answer 1

The correct option is A

$\frac{8}{3}$ sq units

Explanation for the correct option:

Step 1. Draw the graphs of $y=x^2$ and $y=2-x^2$ :

Given that, $y=x^2$ …..(1)

and $y=2-x^2$

$\Rightarrow$ $x^2=-\left(y-2\right)$ …..(2)

By solving equation (1) and (2), we get

$x=-1,1$ and $y=1,1$

Step 2. Required area $=$Area of curve $OABCO$

$=2$Area of curve $OABO$

$$\begin{gathered} =2{\int }_0^{1}ydx \\ =2{\int }_0^1\left[\left(2-x^2\right)-\left(x^2\right)\right]dx \\ =2{\int }_0^1\left(2-2x^2\right)dx \\ =4{\left[x-\frac{x^3}{3}\right]}_0^1 \\ =4\left[1-\frac{1}{3}\right] \\ =\frac{8}{3} \end{gathered}$$

$\therefore$Area bounded by the curves $y=x^2$ and $y=2-x^2$ is $\frac{\mathbf{8}}{\mathbf{3}}$ sq units.

Hence, Option ‘A’ is Correct.