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