Question

Area of the region bounded by two parabolas $y\, =\, x^2$ and $x \,= \, y^2 $ is

• A. $\frac{1}{3}$
• B. $3$
• C. $\frac{1}{4}$
• D. $4$

Answer 1

The Correct Option is A

The intersection point of parabolas $y=x^{2}$ and $x=y^{2}$ is $y=\left(y^{2}\right)^{2}$ $\Rightarrow y=y^{4}$ $\Rightarrow y=0, y=1$ $\Rightarrow x=0, x=1$

So, the intersection point in $O(0,0)$ and $(1,1)$. $\therefore$ Required area $=\int_\limits{0}^{1}\left(y_{2}-y_{1}\right) d x$ $=\int_\limits{0}^{1}\left(\sqrt{x}-x^{2}\right) d x$ $=\left[\frac{x^{3 / 2}}{3 / 2}-\frac{x^{3}}{3}\right]_{0}^{1}=\left[\frac{2}{3} x^{3 / 2}-\frac{x^{3}}{3}\right]_{0}^{1}$ $=\left[\frac{2}{3}(1)-\frac{1}{3}(1)+0-0\right]$ $=\left[\frac{2}{3}-\frac{1}{3}\right]=\frac{1}{3}$ We know that, if parabolas are $y^{2}=4 a x$ and $x^{2}=4by$, then area of bounded region is $\frac{4 a \cdot 4 b}{3}.$ $\therefore$ Required area $=\frac{1 \times 1}{3}=\frac{1}{3}$