5.6. NUMERICAL INTEGRATION
331
(a) Is f increasing or decreasing on [3, 6]? What data tells you?
(b) Is f concave up or concave down on [3, 6]? Why?
(c) Determine the best possible estimate you can for
6
3
f (x) dx, based on the
given information.
3. The rate at which water flows through Table Rock Dam on the White River in Branson,
MO, is measured in thousands of cubic feet per second (TCFS). As engineers open the
floodgates, flow rates are recorded according to the following chart.
seconds, t
0
10
20
30
40
50
60
flow in TCFS, r(t) 2000 2100 2400 3000 3900 5100 6500
(a) 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?
(b) 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 underestimates the exact value of the integral? Why?
(c) Approximate the integral stated in (a) by calculating S n for the largest possible
value of n, based on the given data.
(d) Compute
1
60 S n and
2000+2100+2400+3000+3900+5100+6500
7
. What quantity do both
of these values estimate? Which is a more accurate approximation?