Let [x] denote the greatest integer ≤ x . Consider the function f ( x ) = max x 2 , 1 + [ x ] . Then the value of the integral ∫ 0 2 f ( x ) d x is :

Question

TestHub · Accepted Answer

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

Now when $x \in [0, 1) \Rightarrow f (x) = 1$

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

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

So, from above diagram area can be calculated as,

$A = \int_{0}^{1} 1 . d x + \int_{1}^{\sqrt{2}} 2 d x + \int_{\sqrt{2}}^{2} x^{2} d x$

$\Rightarrow A = 1 + 2 \sqrt{2} - 2 + \frac{8}{3} - \frac{2 \sqrt{2}}{3}$

$\Rightarrow A = \frac{5}{3} + \frac{4 \sqrt{2}}{3}$

Mathematics - Definite Integration Question with Solution | TestHub