Discussion
If \([x]\) is the greatest integer less than or equal to x, then \(\displaystyle\int_{-3}^{3} [x]\, dx =\)
Answer: A
The integral is −3.
\([x]\) is constant on each unit interval, so split the range:
\[ \begin{aligned} \int_{-3}^{3}[x]\,dx &= (-3) + (-2) + (-1) + 0 + 1 + 2 \\ &= -3 \end{aligned} \]
The answer is not 0 because \([x]\) is not an odd function: on [−3, −2) it is −3, but on [2, 3) it is only 2. −6 and −4 miscount the intervals.
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