2023 MIT Integration Bee: Regular Season Q14

Evaluate \(\int_{0}^{100} \lfloor x \rfloor \lceil x \rceil \mathrm{d}x\) .

Note:

• \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\) ; \(\lceil x \rceil\) denotes the smallest integer greater than or equal to \(x\) . For example, \(\lfloor 3.7 \rfloor = 3\), \(\lfloor -2.1 \rfloor = -3\), \(\lceil 3.7 \rceil = 4\), \(\lceil -2.1 \rceil = -2\) .
• You may find the following formulae useful:
  \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2} \quad \quad \quad \sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6} \quad \quad \quad \sum_{k=1}^{n} k^{3} = \frac{n^{2}(n+1)^{2}}{4}\)