Question

If $f\left(x\right)=x^2,g\left(x\right)=2x,0\le x\le 2$ then the value of $I\left(x\right)={\int }_0^{2}max\left(f\right(x),g(x\left)\right)$ is

• A. $\frac{8}{3}$
• B. $\frac{2}{3}$
• C. $\frac{5}{3}$
• D. $\frac{7}{3}$

Answer 1

The correct option is D $\frac{7}{3}$

Let $r\left(x\right)=max\left(f\right(x),g(x\left)\right)$

$\therefore r\left(x\right)=2xxϵ(0,1)$

$andr\left(x\right)=x^{2}xϵ(1,2)$

$\therefore I\left(x\right)={\int }_0^{1}2xdx+{\int }_1^2{x}^{2}dx$

$\therefore I\left(x\right)=[x^2{]}_0^1+\frac{[x^3{]}_1^2}{3}=(1-0)+\frac{8-1}{3}=\frac{7}{3}$

Hence, option D is correct.