Lebesgue integral of function given by series

Question

Calculate the integral of function $f(x):\mathbb{R} \rightarrow \mathbb{R}$, where $f(x) =\displaystyle \sum_{k=0}^{120} \lfloor x\rfloor \cdot \chi_{(k-1,k)} (x)$.

How to calculate this?

Answer 1

We have $\int_{\mathbb{R}}\lfloor x\rfloor \chi_{(k-1,k)}(x) dx=\int_{k-1}^k\lfloor x\rfloor dx$ $=\int_{k-1}^k (k-1) dx$

It remains to calculate $\sum_{k=0}^{120} (k-1)=7139$