Question

The table (supplied by San Diego Gas and Electric) gives the power consumption \(P\) in megawatts in San Diego County from midnight to\({\bf{\;6:00AM}}\)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.)

Short answer

Energy used during time period is \(10,177\)megawatt.

Step-by-step solution

Simpson's Rule

By the formula\(\int_a^b f (x)dx \approx {S_n} = \frac{{\Delta x}}{3}\left( {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) + 4f\left( {{x_3}} \right) + L + 2f\left( {{x_{n - 2}}} \right) + 4f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right)\)

Where\(x\)is even and\(\Delta x = \frac{{b - a}}{n}\).

If\({E_s}\)is the error involved in the Simpson's rule then\(\left| {{E_s}} \right| \le \frac{{k{{(b - a)}^5}}}{{180{n^4}}}\).

Computation

Given a table of data for power & time.

We know that power is derivative of energy so we can estimate energy used using Simpson’s rule.

Since the time is changing by 30 seconds so here \(\Delta x = 30\)& there are 12 sub-intervals. Use Simpson's Rule to calculate\({S_{12}}\).

\(\begin{aligned}{c}{S_{12}} = \frac{{6 - 0}}{{3(12)}}(P(0:00) + 4f(0:30) + 2(1:00) \ldots + 4f(5:30) + f(6:00))\\ = \frac{6}{{36}}(1814 + 4(1735) + 2(1686) + 4(1646) + \ldots + 4(1886) + 2052)\\ \approx 10,177\end{aligned}\)

Therefore, \(10,177\) megawatt-hours were used from midnight to \(6:00{\rm{AM}}\).