Let [x] denote the greatest integer ≤ x . Consider the function f (...

Plotting the diagram of the given function $f ( x ) = max\left\{{x}^{2}, 1 + [ x ]\right\}$ we get,

Now when $x \in [ 0, 1 ) ⇒ f\left(x\right) = 1$

When $x \in [ 1, \sqrt{2} ) ⇒ f\left(x\right) = 2$

And when $x \in \left[\sqrt{2}, 2\right] ⇒ f\left(x\right) = {x}^{2}$

So, from above diagram area can be calculated as,

$A = {∫}_{0}^{1}1 . dx + {∫}_{1}^{\sqrt{2}}2dx + {∫}_{\sqrt{2}}^{2}{x}^{2}dx$

$⇒ A = 1 + 2\sqrt{2} - 2 + \dfrac{8}{3} - \dfrac{2\sqrt{2}}{3}$

$⇒ A = \dfrac{5}{3} + \dfrac{4\sqrt{2}}{3}$