300 followers problem

$We \ have, \ f(x)\quad =\quad max\left\{ { x }^{ 2 },{ \left( 1-x \right) }^{ 2 },2x(1-x) \right\}$

Considering only the above max region. So, f(x) can be written as,

$f(x)=\left\{ \begin{matrix} { (1-x) }^{ 2 }\quad ,\quad \quad \quad for\quad 0\le x\le \frac { 1 }{ 3 } \\ 2x(1-x)\quad ,\quad \quad \quad for\quad \frac { 1 }{ 3 } \le x\le \frac { 2 }{ 3 } \\ { x }^{ 2 }\quad ,\quad \quad \quad \quad \quad for\quad \frac { 2 }{ 3 } \le x\le 1 \end{matrix} \right\}$

$\therefore \ Area\ Bounded =\int _{ 0 }^{ \frac { 1 }{ 3 } }{ { \left( 1-x \right) }^{ 2 } } dx \ +\int _{ \frac { 1 }{ 3 } }^{ \frac { 2 }{ 3 } }{ 2x(1-x) } dx \ +\int _{ \frac { 2 }{ 3 } }^{ 1 }{ \left( { x }^{ 2 } \right) } dx$

$= \large{\boxed{\frac{17}{27} sq. \ units}}$

$\therefore \quad \quad \quad a\quad +\quad b\quad =\quad \boxed{44}$