Consider the functions f , g : ℝ → ℝ defined by f x = x 2 + 5 12 and g x = 2 1 - 4 x 3 , x…

Plotting the diagram of the given function we have,

Now by diagram we can see twice of $A_1$ and $A_2$ will be the required area,

So, required area $=2·\left(A_1+A_2\right)$

$\Rightarrow \alpha =\left({\int }_0^{\frac{1}{2}}\left(x^2+\frac{5}{12}\right)dx+\frac{1}{2}\left(\frac{3}{4}-\frac{1}{2}\right)·\frac{2}{3}\right)\times 2$

$\Rightarrow \alpha =\left({\left.\left(\frac{x^3}{3}+\frac{5x}{12}\right)\right|}_0^{\frac{1}{2}}+\frac{1}{2}\cdot \frac{1}{4}\cdot \frac{2}{3}\right)\times 2$

$\Rightarrow \alpha =\left(\frac{1}{4}+\frac{1}{12}\right)\times 2$

$\Rightarrow \alpha =\frac{2}{3}\Rightarrow 9\alpha =6$