Let [x] denote the greatest integer x.

Question

Let $[x]$ denote the greatest integer $\le x$. Consider $f(x)=\max\{x^2,\,1+[x]\}$. Then the value of $\displaystyle\int_0^2 f(x)\,dx$ is:

Accepted Answer

On $[0,1)$: $1+[x]=1 \ge x^2$, so $f(x)=1$. On $[1,\sqrt2)$: $1+[x]=2\ge x^2$, so $f(x)=2$. On $[\sqrt2,2]$: $x^2\ge2 = 1+[x]$, so $f(x)=x^2$.

$$\int_0^2f(x)\,dx = \int_0^1 1\,dx+\int_1^{\sqrt2}2\,dx+\int_{\sqrt2}^2x^2\,dx = 1+2(\sqrt2-1)+\left(\frac83-\frac{2\sqrt2}3\right) = \frac{5+4\sqrt2}3$$

Solutions · 1