Estimate the energy used.

Question

The table gives the power consumption P in megawatts in a county from midnight to 6:00 AM on a day in December. Use Simpson's Rule to estimate the energy used during that time period. (Use the fact that power is the derivative of energy. Round your answer to one decimal place.)

__________ megawatt-hours

t	P	t	P
0:00	1818	3:30	1617
0:30	1733	4:00	1627
1:00	1682	4:30	1668
1:30	1646	5:00	1743
2:00	1639	5:30	1888
2:30	1609	6:00	2058
3:00	1608

Answer 1

Simpson's rule can be used to estimate the area under a curve when there is an even number of sub-intervals with at least 4. Here you have 12 sub intervals and they are even. The rule states:

Area is approximately = Δx/3(f(x₀ + 4f(x₁₎ + 2f(x₂) ............4*f(x_n-1) + f(x_n))

I like to think of it as the interval (here 0.5) over 3 times the sum of THE FIRST, THE LAST and a 4/2/4 combo of all the middles.

Here Δx/3 = 0.16667 (or 0.5/3). The first is 1818, the last is 2058. So here you go:

0.16667(1818 + 4(1733)+2(1682) + 4(1646) + 2(1639)..etc plus 4(1888) + 2058). You do the math.

Just start with the first, then 4 of the next, 2 of the next, 4 of the next..until you get to the second to the last (which will be times 4 if you have an even number like described above) and then add the last. Multiply the entire thing times the dx/3 and you get the answer.

I came up with 10186.33 watts.