Exercise 7 — Active Calculus

The rate at which water flows through Table Rock Dam on the White River in Branson, MO, is measured in cubic feet per second (CFS). As engineers open the floodgates, flow rates are recorded according to the following chart.

Water flow data.
seconds, $t$	$0$	$10$	$20$	$30$	$40$	$50$	$60$
flow in CFS, $r(t)$	$2000$	$2100$	$2400$	$3000$	$3900$	$5100$	$6500$

What definite integral measures the total volume of water to flow through the dam in the 60 second time period provided by the table above?
Use the given data to calculate $M_n$ for the largest possible value of $n$ to approximate the integral you stated in (a). Do you think $M_n$ over- or under-estimates the exact value of the integral? Why?
Approximate the integral stated in (a) by calculating $S_n$ for the largest possible value of $n$ , based on the given data.
Compute $\frac{1}{60} S_n$ and $\frac{2000+2100+2400+3000+3900+5100+6500}{7}$ . What quantity do both of these values estimate? Which is a more accurate approximation?

Boelkins, Active Calculus, §5.6 Numerical integration, Exercise 7

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