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TRAP: $\frac{LRR + RRR}{2}$
Each of these estimates corresponds to computing a rectangle height (a "representative" or average speed for that interval), and then multiplying by the total time. Hence we're using the "dirt" formula, and computing the average (the "area under the curve", even though there's no curve!) in several different ways.
If we take the limit as $n \to \infty$ (that is, drive $\Delta t=\frac{b}{n} \to 0$), then we get the exact distance (which is equivalent to an area), and represent it with this integral notation: \[ p(b)-p(0)=\int_{0}^{b}p'(t)dt \]
This is called a definite integral, because the limits of integration (0 and $b$) are fixed (definite). In the near future we will be relaxing that, and allowing "indefinite" limits.
Generalizations?
where by $\bar{f}\mbox{}$ we mean an average value of $f$.
and then we give a name to this area:
where P is called a partition of the interval $[a,b]$:
and where C is a set of intermediate points $C={c_1,\ldots,c_N}$ such that
where
If $f$ is continuous on $[a,b]$, then