In this video I go over a very useful application on using Simpson's Rule to approximate a function that has no actual function given. Instead in this example, only a chart is given of the total megabits of data transferred from the United States to the Swiss academic and research network, SWITCH, on February 10, 1998. Since the chart contains data that is smooth without random bumps in the data, we can apply the Simpson's Rule by simply estimating data points for the summation. Also in this video I discuss a bit about the difference between megabits and megabytes so make sure to watch this video!
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Approximate Integration: Example 5: Simpson's Rule
In many applications of calculus we need to evaluate an integral even if no explicit formula is known for y as a function of x. A function may be given graphically or as a table of values of collected data. If there is evidence that the values are not changing rapidly, then the Trapezoidal Rule or Simpson's Rule can be used to find an approximate value for the integral of y with respect to x:
The figure below shows data traffic on the link from the United States to SWITCH, the Swiss academic and research network, on February 10, 1998. D(t) is the data throughput, measured in megabits per second (Mb/s or Mbps). Use Simpson's Rule to estimate the total amount of data transmitted on the link up to noon on that day.
Note: A megabit (Mb) should not be confused with a megabyte (MB)
- Mb is used for download speeds
- MB is used for file size
- 1 MB = 8 Mb (same with 1 B = 8 b , 1 GB = 8 Gb, etc.)
- Thus a download speed of 8 Mb per second (Mbps) downloads 1 MB every second.
Solution:
Because we want the units to be consistent and D(t) is measured in Mb/s, we convert the units for t from hours to seconds.
Since D(t) is a rate of change of Mb (i.e. Mb/s), the integral or area underneath the curve simply gives the net change (learn more from my video on the Net Change Theorem)
- In our case, D(t) is always positive so the area under the curve is just the total Mb downloaded
Letting A(t) be the total Mb transmitted by time t, we have:
Now we need to estimate the values of D(t) at hourly intervals from the graph and compile them in a table:
- Note that since we are only asked to find the total data transmitted by noon, we only need to estimate the first 12 hours of data.
Using Simpson's Rule we have:
