Question

The total area bounded by the closed region of $y^2+x^4=x^2$ in square units is

• A. $\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $\frac{8}{3}$
• D. $2$

Answer 1

The correct option is A $\frac{2}{3}$

Let

$y^2=x^2\left(1-x^2\right)$

$=2\underset{0}{\overset{1}{\int }}x\sqrt{1-x^2}dx$

$=-\underset{0}{\overset{1}{\int }}-2x\sqrt{1-x^2}dx$

$=-\frac{2}{3}{\left[{\left(1-x^2\right)}^{\frac{3}{2}}\right]}_0^1$

$=\frac{-2}{3}\left[0-1\right]=\frac{2}{3}$

Hence, the option $\left(A\right)$ is the correct answer.