question_answer

The area of the region bounded by \[y=2x-{{x}^{2}}\]and the x -axis is

A)  \[\frac{8}{3}sq\,unit\]                  

B)  \[\frac{4}{3}sq\,unit\]

C)  \[\frac{7}{3}sq\,unit\]                  

D)  \[\frac{2}{3}sq\,unit\]

Correct Answer: B

Solution :

Given curve is \[y=2x-{{x}^{2}}\]                 or            \[{{(x-1)}^{2}}=-(y-1)\]                                 The curve cut the x- axis at \[(0,0)\] and \[(2,0)\]. \[\therefore \] Required area \[=\int_{0}^{2}{y\,dx}\]                 \[\int_{0}^{2}{(2x-{{x}^{2}})dx=\left[ {{x}^{2}}-\frac{{{x}^{3}}}{3} \right]_{0}^{2}}\]                 \[=4-\frac{8}{3}=\frac{4}{3}sq\,unit\]