question_answer

The area bounded by the curve\[y=2x-{{x}^{2}}\]and the line\[y=-x\]is

A)  \[\frac{3}{2}\]sq units     

B)  \[\frac{9}{3}\]sq units

C)  \[\frac{9}{2}\]sq units    

D)  None of these

Correct Answer: C

Solution :

Given curve \[y=2x-{{x}^{2}}\] \[\Rightarrow \] \[{{(x-1)}^{2}}=-(y-1)\] and line \[y=-x\]. The point of intersection are (0, 0) and\[(-3,3)\]. \[\therefore \]Required area\[=\int_{0}^{3}{[(2x-{{x}^{2}})-(-x)]}\,dx\] \[=\int_{0}^{3}{(3x-{{x}^{2}})}\,dx\] \[=\left( \frac{3{{x}^{2}}}{2}-\frac{{{x}^{3}}}{3} \right)_{0}^{3}\] \[=\frac{9}{2}\]sq units