Expert: Scotto - 1/27/2010

Question
The table below gives the power consumption in megawatts in San Diego Country from midnight to 6:00am on December 8, 1999. Use Simpson's Rule to estimate the energy used during that time period. (Note that power is the derivative of energy.)

t          P
0:00       1814
0:30       1735
1:00       1686
1:30       1646
2:00       1637
2:30       1609
3:00       1604
3:30       1611
4:00       1621
4:30       1666
5:00       1745
5:30       1886
6:00       2052


The time here is in hours. To use the Simpson's Rule, i need to have an even amount of intervals, here i counted 13. I got this problem out of a book, can you please explain this to me step by step, i am really not that good at this.

Answer
There are 13 points, which make 12 intervals.

There is one interval from 0:00 to 1:00, another from 1:00 to 2:00, etc, and the last is from 5:00 to 6:00, a total of 6 in all with the midpoint in each one of them.

I believe that Simpson's Rule says to take [f(a) + 4f((a+b)/2) + f(b)](b-a)/6 for each interval.
Here, the points where x is (a+b)/2 is at the halfhour in between each hour.
The points are on the hour.

So Simpson's Rule would be to add the values at 0:00 and 6:00,
take twice the sum of the values at 1:00, 2:00, 3:00, 4:00, and 5:00,
and then 4 times the sum of the values on every time ending in ":30".
The reason that the values at 1:00, 2:00, 3:00, 4:00 and 5:00 are counted twic
is that they are at the left point of one interval and at the right point of the other.

The amount to multiply by is the differences between each successive pair of 7 points,
which are the hours, and it would be 1.  Divide this answer by 6.