question_answer

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

A) \[\frac{9}{2}sq\,\,unit\]               

B) \[\frac{43}{6}sq\,\,unit\]

C) \[\frac{35}{6}sq\,\,unit\]                             

D)  none of these

Correct Answer: A

Solution :

Required area\[=\int_{0}^{3}{({{y}_{1}}-{{y}_{2}})}\,dx\]                           \[=\int_{0}^{3}{(2x-{{x}^{2}})}-(-x)dx\]                 \[=\int_{0}^{3}{(3x-{{x}^{2}})}\,\,dx\]                 \[=\left[ \frac{3{{x}^{2}}}{2}-\frac{{{x}^{3}}}{3} \right]_{0}^{3}\]                 \[=\frac{27}{2}-9\]                 \[=\frac{9}{2}sq\,\,unit\]