Let y=[x]. Now let's find out [0.5] as an example. The largest integer for x≤0.5 is 0, so y=0 when x=0.5. When 1≤x<2, y=1. In other words, whatever integer precedes the decimal point is [x].
If y=[x], we can tabulate values of y for all values of x between 1 and 7:
| x |
y |
| 1≤x<2 |
1 |
| 2≤x<3 |
2 |
| 3≤x<4 |
3 |
| 4≤x<5 |
4 |
| 5≤x<6 |
5 |
| 6≤x<7 |
6 |
| x=7 |
7 |
When x=7 we have a single point (7,7). The integral of [x] is just the area under the graph of y=[x]. The graph is not continuous because it consists of a number of discrete steps, as in a staircase with no risers. However, the area beneath the steps can be calculated. Under the first step, the area is 1 square unit, under the second step it's 2 square units, and so on. But when we get to x=7 we only have a single point so there is no area (because there's no width), or area→0 in the limit. The total area is 1+2+3+4+5+6=21 square units. The individual areas are rectangles with a base length limit → 1.