Integrate floor

The floor function is a step function. For the integrand $\lfloor 2x + 3 \rfloor$ , where $x \in (-3,3)$ , it is shown in the graph above. It is a stair or $y$ ranging from $-3$ to $8$ , each step a height of $1$ and width of $\frac 12$ . The integral of floor function is the area under the stair steps. In equation, we have:

$\begin{aligned} \int_{-3}^3 \lfloor 2x+3\rfloor & = \frac 12 \sum_{k=-3}^3 k \\ & = \frac 12 \left(-\sum_{k=1}^3 k + \sum_{k=1}^8 k \right) & \small \blue{\text{Steps }-3, -2, -1 \text{ cancel with steps }+1, + 2, +3} \\ & = \frac 12 \sum_{k=4}^8 k & \small \blue{\text{Remaining 5 steps, 4 to 8 (as shaded)}} \\ & = \frac 12 \times \frac {5(4+8)}2 = \boxed {15} \end{aligned}$

The integrand is $\lfloor {2x+3}\rfloor=\lfloor {2x}\rfloor +3$ . So the value of the integral is $3(3+3)+0.5(-6-5-4-3-2-1+1+2+3+4+5)=18-3=\boxed {15}$ .