If the upper and lower integrals are equal, we say that $f$ is Riemann-integrable on $[a, b]$.
We use $\sup$ and $\inf$ to define the area under a curve.
We associate to each partition $P = \{x_0, \dots, x_n\}$ of $[a,b]$ and to each curve $\gamma$ on $[a, b]$ the number $$\Lambda(P, \gamma)=\sum_{i=1}^{n} | \gamma(x_i)-\gamma(x_{i-1})|.$$
The length of $\gamma$ is defined as $$\sup \Lambda(P, \gamma).$$
We use only $\sup$ to define the length of a curve.
Why?
Thinking about nicely smooth curves for a bit, this is the fact that the shortest distance between two points is a straight line. So, your straight-line approximation by meshing should always be less than or (with very small probability) equal to the true length of the curve. So, you definitely don't want the infimum. Intuitively, as your mesh becomes finer the approximation should get better, but the triangle inequality also says the estimate should get larger than the previous. (Consider subdividing $[x_i, x_{i+1}]$ into halves and drawing the new approximation.) So, the supremum is the way to go.