Chapter 1.2, Problem 134E

Suppose that for each i such that $1\le i\le N$ one has ${\displaystyle {\int }_{i-1}^{i}f\left(t\right)dt=i}$ . Show that ${\displaystyle {\int }_0^{N}f\left(t\right)dt=\frac{N\left(N+1\right)}{2}}$ .