question_answer

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

A) \[\frac{1}{9}sq\,\,unit\]               

B) \[\frac{1}{3}sq\,\,unit\]

C) \[\frac{1}{\sqrt{3}}sq\,\,unit\]                  

D) \[\frac{2}{3}sq\,\,unit\]

Correct Answer: B

Solution :

Given curves\[y={{x}^{2}},\,\,x={{y}^{2}}\] \[\therefore \]Point of intersections                 \[x={{({{x}^{2}})}^{2}}\Rightarrow {{x}^{4}}-x=0\] \[\Rightarrow \]                                \[x({{x}^{3}}-1)=0\] \[\therefore \]                                  \[x=0,\,\,1\] For         \[x=0,\,\,y=0;\,\,x=1,\,\,y=1\] Area of shaded region                 \[=\int_{0}^{1}{{{x}^{1/2}}}dx-\int_{0}^{1}{{{x}^{2}}dx}\]                 \[=\left[ \frac{2{{x}^{3/2}}}{3} \right]_{0}^{1}-\left[ \frac{1}{3}{{x}^{3}} \right]_{0}^{1}\]                 \[=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}sq\,\,unit\] \[\therefore \]Required area\[=\frac{1}{3}sq\,\,unit\]