Evaluate $\int_{0}^{100} \lfloor x \rfloor \lceil x \rceil \mathrm{d}x$ . Note: \(\lfloor x \rfloo

Question

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, $\lfloor 3.7 \rfloor = 3$, $\lfloor -2.1 \rfloor = -3$, $\lceil 3.7 \rceil = 4$, $\lfloor -2.1 \rfloor = -2$ . You may find the following formulae useful: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2} \quad \quad \quad \sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6} \quad \quad \quad \sum_{k=1}^{n} k^{3} = \frac{n^{2}(n+1)^{2}}{4}$

Accepted Answer

$\frac{100^{4} - 100^{2}}{4}$

2023 MIT Integration Bee: Regular Season Q14

2023 MIT Integration Bee: Regular Season Q14