question_answer

    The area enclosed between the parabola \[y={{x}^{2}}-x+2\]and the line\[y=x+2\]in sq unit equals

A)  8/3                                       

B)  1/3

C)  2/3                                       

D)  4/3

Correct Answer: D

Solution :

                Given equation of parabola is                 \[y={{x}^{2}}-x+2\] Or           \[{{\left( x-\frac{1}{2} \right)}^{2}}=y-\frac{7}{4}\] and equation of line is \[y=x+2\] \[\therefore \]Required area \[=\int_{0}^{2}{[(x+2)-({{x}^{2}}-x+2)]}\,dx\] \[=\int_{0}^{2}{(-{{x}^{2}}+2x)}\,dx\]                 \[=\left[ -\frac{{{x}^{3}}}{3}+{{x}^{2}} \right]_{0}^{2}=-\frac{8}{3}+4\] \[=\frac{4}{3}\]sq unit