Definite Integration Question 9
Question 9 - 29 January - Shift 1
Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x)=\max \lbrace x^{2}, 1+[x] \rbrace $. Then the value of the integral $\int_0^{2} f(x) d x$ is :
(1) $\frac{5+4 \sqrt{2}}{3}$
(2) $\frac{8+4 \sqrt{2}}{3}$
(3) $\frac{1+5 \sqrt{2}}{3}$
(4) $\frac{4+5 \sqrt{2}}{3}$
Show Answer
Answer: (1)
Solution:
Formula: Greatest Integer Function, Properties of definite integral
$\begin{aligned} & A=\int_0^1 1 \cdot d x+\int_1^{\sqrt{2}} 2 d x+\int_{\sqrt{2}}^2 x^2 d x \\ & =1+2 \sqrt{2}-2+\frac{8}{3}-\frac{2 \sqrt{2}}{3} \\ & =\frac{5}{3}+\frac{4 \sqrt{2}}{3}\end{aligned}$
