Problem 2E

Suppose $f$ is a continuous positive decreasing function for $x \geqslant 1$ and $a_{n}=f(n) .$ By drawing a picture, rank the following three quantities in increasing order:

$$\int_{1}^{6} f(x) d x \quad \sum_{i=1}^{5} a_{i} \quad \sum_{i=2}^{6} a_{i}$$

Step-by-Step Solution