question_answer

Area between curves\[y={{x}^{2}},\,\,x={{y}^{2}}\]is

A)  \[\frac{1}{9}\]                       

B)  \[\frac{1}{3}\]

C)  \[\frac{1}{\sqrt{3}}\]                                    

D)  \[\frac{2}{3}\]

Correct Answer: B

Solution :

Solving, y = x2 and x = y2 \[y={{y}^{4}}\]or \[y({{y}^{3}}-1)=0\Rightarrow y=0\]or y = 1 \[\therefore \]Point of intersection are (0,0) &(1,1) To find the shaded area, \[A=\int_{0}^{1}{(\sqrt{x}-{{x}^{2}})dx}\] \[=\frac{2}{3}\left[ {{x}^{3/2}} \right]_{0}^{1}-\left[ \frac{{{x}^{3}}}{3} \right]_{0}^{1}=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}\]