The table (supplied by San Diego Gas and Electric) gives the power consumption in megawatts in San Diego County from midnight to 6:00 a.m. on December 8, 1999. Use Simpson's Rule to estimate the energy used during that time period, giving your answer correct to two decimal places. Use the fact that power is the derivative of energy and that the units of energy are megawatt-hours.

Time	P (mw)
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
Energy used ≈
 
 megawatt-hours

We will apply **Simpson’s Rule** to estimate the energy used during the time period. Simpson's Rule is defined as:

\[
\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + f(x_n)\right],
\]

where:
- \( n \) (the number of subintervals) must be even,
- \( \Delta x = \frac{b - a}{n} \),
- and the coefficients alternate between \( 4 \) and \( 2 \), with \( 1 \) at both ends.

---

### Step 1: Key Information
- Time is from \( 0:00 \) to \( 6:00 \), so \( a = 0 \) and \( b = 6 \) hours.
- There are 13 data points (at 30-minute intervals), so \( n = 12 \) subintervals (even as required).
- \( \Delta x = \frac{6 - 0}{12} = 0.5 \) hours.

---

### Step 2: Assign Power Values to \( P(x) \)
The power values correspond to \( f(x) \), as follows:
\[
P(x) = [1814, 1735, 1686, 1646, 1637, 1609, 1604, 1611, 1621, 1666, 1745, 1886, 2052].
\]

---

### Step 3: Apply Simpson’s Rule
Using the formula:
\[
\int_{0}^{6} P(t) \, dt \approx \frac{\Delta x}{3} \left[P_0 + 4P_1 + 2P_2 + 4P_3 + \cdots + 2P_{n-2} + 4P_{n-1} + P_n\right],
\]
we substitute the power values:
\[
\int_{0}^{6} P(t) \, dt \approx \frac{0.5}{3} \Big[1814 + 4(1735) + 2(1686) + 4(1646) + 2(1637) + 4(1609) + 2(1604) + 4(1611) + 2(1621) + 4(1666) + 2(1745) + 4(1886) + 2052\Big].
\]

---

### Step 4: Simplify the Expression
Evaluate each term:
\[
\begin{aligned}
\int_{0}^{6} P(t) \, dt &\approx \frac{0.5}{3} \Big[1814 + 6940 + 3372 + 6584 + 3274 + 6436 + 3208 + 6444 + 3242 + 6664 + 3490 + 7544 + 2052\Big] \\
&\approx \frac{0.5}{3} \Big[59764\Big].
\]

Now compute:
\[
\int_{0}^{6} P(t) \, dt \approx \frac{0.5}{3} \times 59764 = \frac{29882}{3} \approx 9960.67 \, \text{megawatt-hours}.
\]

---

### Final Answer:
The energy used during this time period is approximately:
\[
\boxed{9960.67 \, \text{megawatt-hours}}.
\]

Would you like more explanation or a breakdown of any step? 😊

---

### Questions:
1. What is Simpson’s Rule, and why is it accurate for numerical integration?
2. How does the step size (\( \Delta x \)) affect the accuracy of Simpson’s Rule?
3. What is the physical relationship between power and energy in this problem?
4. How would the result change if \( n \) were not even? Could we still apply Simpson’s Rule?
5. How can we use trapezoidal rule to solve this problem, and how does it compare to Simpson’s Rule?

---

### Tip:
Simpson’s Rule requires the function to be smooth and well-behaved; it’s particularly accurate for functions resembling parabolas over small intervals.

The table (supplied by San Diego Gas and Electric) gives the power consumption in megawatts in San Diego County from midnight to 6:00 a.m. on December 8, 1999. Use Simpson's Rule to estimate the energy used during that time period, giving your answer correct to two decimal places. Use the fact that power is the derivative of energy and that the units of energy are megawatt-hours.

Estimate the energy used in San Diego County from midnight to 6:00 a.m. on December 8, 1999, using Simpson's Rule. Learn how to apply numerical integration to power data and understand the relationship between power and energy. Answer: 9960.67 megawatt-hours.

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