Question

Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x)=\max \left\{x^2, 1+[x]\right\}$. Then the value of the integral $\int\limits_0^2 f(x) d x$ is

• A. $\frac{5+4 \sqrt{2}}{3}$
• B. $\frac{8+4 \sqrt{2}}{3}$
• C. $\frac{1+5 \sqrt{2}}{3}$
• D. $\frac{4+5 \sqrt{2}}{3}$

Answer 1

Answer: (A)

$ A=\int\limits_0^1 1 \cdot d x+\int\limits_1^{\sqrt{2}} 2 d x+\int\limits_{\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}$