Thread: Word problem and reimann sums.

We have a creek that is 20 ft wide with depth measurements recorded at two foot intervals.
Here is a table (table 1):

-Distance from bank (in): 24 -depth(in): 5.5
- Distance from bank (in): 48 -depth(in): 7.75
-Distance from bank (in): 72 -depth(in): 9.5
-Distance from bank (in): 96 -depth(in): 11
-Distance from bank (in): 120 -depth(in): 13.5
-Distance from bank (in): 144 depth(in): 16
-Distance from bank (in): 163 -depth(in): 13.75
-Distance from bank (in): 192 -depth(in): 12
-Distance from bank (in): 216 -depth(in): 6
-Distance from bank (in): 240 -depth(in): 0

The steam velocity was also measured.
Here is a table (table 2):

-Distance from bank (in): 24 -velocity (in/sec): 3.84
-Distance from bank (in): 48 -velocity (in/sec): 9
-Distance from bank (in): 72 -velocity (in/sec): 10.8
-Distance from bank (in): 96 -velocity (in/sec): 18
-Distance from bank (in): 120 -velocity (in/sec): 21.12
-Distance from bank (in): 144 -velocity (in/sec): 30.48
-Distance from bank (in): 163 -velocity (in/sec): 19.2
-Distance from bank (in): 192 -velocity (in/sec): 5.28
-Distance from bank (in): 216 -velocity (in/sec): 0.96
-Distance from bank (in): 240 -velocity (in/sec): 0

In part A, we were told to graph the stream showing the depth as a function of distance. This was easy.

In part B, it asks, “Use the idea of Riemann Sums to find an upper and a lower bound for the area of the cross-section of the stream at this site. (You may use trapezoids or some other shape instead of rectangles if you think they will give more accurate results.)”
I used the “Trapezoidal Rule” to find the total area being 2280 square in. How do I find the upper and lower bound? Are these error bounds they are talking about?


In part C, is asks, “Explain how to use the velocity data in conjunction with the rectangles (or trapezoids or something else) from your Riemann Sums to get upper and lower bounds for the total stream flow at the site and explain how you know they are upper and lower bounds.”
How do I find the upper and lower bounds for this?



Sorry this is a loaded question, but thanks for reading!

Originally Posted by NeedDirection

We have a creek that is 20 ft wide with depth measurements recorded at two foot intervals.
Here is a table (table 1):

-Distance from bank (in): 24 -depth(in): 5.5
- Distance from bank (in): 48 -depth(in): 7.75
-Distance from bank (in): 72 -depth(in): 9.5
-Distance from bank (in): 96 -depth(in): 11
-Distance from bank (in): 120 -depth(in): 13.5
-Distance from bank (in): 144 depth(in): 16
-Distance from bank (in): 163 -depth(in): 13.75
-Distance from bank (in): 192 -depth(in): 12
-Distance from bank (in): 216 -depth(in): 6
-Distance from bank (in): 240 -depth(in): 0

The steam velocity was also measured.
Here is a table (table 2):

-Distance from bank (in): 24 -velocity (in/sec): 3.84
-Distance from bank (in): 48 -velocity (in/sec): 9
-Distance from bank (in): 72 -velocity (in/sec): 10.8
-Distance from bank (in): 96 -velocity (in/sec): 18
-Distance from bank (in): 120 -velocity (in/sec): 21.12
-Distance from bank (in): 144 -velocity (in/sec): 30.48
-Distance from bank (in): 163 -velocity (in/sec): 19.2
-Distance from bank (in): 192 -velocity (in/sec): 5.28
-Distance from bank (in): 216 -velocity (in/sec): 0.96
-Distance from bank (in): 240 -velocity (in/sec): 0

In part A, we were told to graph the stream showing the depth as a function of distance. This was easy.

In part B, it asks, “Use the idea of Riemann Sums to find an upper and a lower bound for the area of the cross-section of the stream at this site. (You may use trapezoids or some other shape instead of rectangles if you think they will give more accurate results.)”
I used the “Trapezoidal Rule” to find the total area being 2280 square in. How do I find the upper and lower bound? Are these error bounds they are talking about?


In part C, is asks, “Explain how to use the velocity data in conjunction with the rectangles (or trapezoids or something else) from your Riemann Sums to get upper and lower bounds for the total stream flow at the site and explain how you know they are upper and lower bounds.”
How do I find the upper and lower bounds for this?



Sorry this is a loaded question, but thanks for reading!

If we assume that the depth varies sufficiently smoothly with distance from the bank you may take it that the depth in an interval is greater than the smaller of the intervals end values and less than the greater.

So between 0 and 24 inches from the bank the depth is greater than 0 and less than 5.5 inches, and so on for the other intervals between the spot measuremants.

Then using the lower values and upper values respectivly in the Riemann sum you will get an upper and lowwer bound for the integral.

CB