Question

The area enclosed between the parabola $y=x^2-x+2$ and the line $y=x+2$ in sq unit is equal to

• A. $\frac{8}{3}$
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{4}{3}$

Answer 1

The correct option is D $\frac{4}{3}$

Given, equation of parabola is
$y=x^2-x+2$ ....(i)
$\Rightarrow {(x-\frac{1}{2})}^2=y-\frac{7}{4}$
and equation of line is
$y=x+2$ ...(ii)
On solving equations (i) and (ii), we get
$x^2-x+2=x+2$
$x^2-2x=0$
$x(x-2)=0,x=0,2$
Therefore, required area will be
$={\int }_0^2[(x+2)-(x^2-x+2)]dx$
$={\int }_0^2(-x^2+2x)dx$
$={[-\frac{x^3}{3}+x^2]}_0^2=-\frac{8}{3}+4=\frac{4}{3}$ sq units