The graph shows a​ city's daily rate of use of electricity​ (in millions of​ kilowatts). Estimate the total daily usage of electricity by summing the area of rectangles.  Use the left​ endpoints, then the right​ endpoints, then give the average of those results as the answer.  Let the width of each rectangle be 2 hours.
Midnight
4 A.M.
8 A.M.
Noon
4 P.M.
8 P.M.
Midnight
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4
8
12
16
20
24
28
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Millions of kilowatt-hours



A coordinate system has a horizontal axis labeled with hours from Midnight to Midnight, in increments of 4 hours and a vertical axis labeled Millions of kilowatt-hours from 0 to 24 in increments of 4. A smooth curve rises from right to left, passing through the points with the following y-coordinates, with the hours listed first: Midnight, 11.29; 4 A.M., 12.62; 8 A.M., 16.62; Noon, 23.29, to a maximum at 4 P.M, 29.15, and falls through the following points: 8 P.M., 20.49; Midnight, 13.48. All coordinates are approximate.
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Part 1
Begin by finding the estimate of the total energy used on the day shown in the​ graph, by summing rectangles using left endpoints.  The width of each interval is 2 hours.

To estimate the total daily electricity usage using the left endpoints, we can break the 24-hour period into 2-hour intervals and sum the areas of rectangles formed by these intervals. 

### Given Data
The points you provided for the electricity usage are:

- Midnight (0:00) - 11.29 million kWh
- 4 A.M. (4:00) - 12.62 million kWh
- 8 A.M. (8:00) - 16.62 million kWh
- Noon (12:00) - 23.29 million kWh
- 4 P.M. (16:00) - 29.15 million kWh
- 8 P.M. (20:00) - 20.49 million kWh
- Midnight (24:00) - 13.48 million kWh

### Left Endpoint Calculation
Using the left endpoints, we take the height of the function at the beginning of each interval:

1. **Midnight to 4 A.M. (0-4)**:
   - Height = 11.29 million kWh
   - Area = \( 11.29 \times 2 = 22.58 \) million kWh

2. **4 A.M. to 8 A.M. (4-8)**:
   - Height = 12.62 million kWh
   - Area = \( 12.62 \times 2 = 25.24 \) million kWh

3. **8 A.M. to Noon (8-12)**:
   - Height = 16.62 million kWh
   - Area = \( 16.62 \times 2 = 33.24 \) million kWh

4. **Noon to 4 P.M. (12-16)**:
   - Height = 23.29 million kWh
   - Area = \( 23.29 \times 2 = 46.58 \) million kWh

5. **4 P.M. to 8 P.M. (16-20)**:
   - Height = 29.15 million kWh
   - Area = \( 29.15 \times 2 = 58.30 \) million kWh

6. **8 P.M. to Midnight (20-24)**:
   - Height = 20.49 million kWh
   - Area = \( 20.49 \times 2 = 40.98 \) million kWh

### Total Area (Left Endpoint)
Now we can sum these areas:

\[
\text{Total Area} = 22.58 + 25.24 + 33.24 + 46.58 + 58.30 + 40.98 = 226.62 \text{ million kWh}
\]

### Right Endpoint Calculation
Now, we calculate using the right endpoints. The heights will be taken from the end of each interval:

1. **Midnight to 4 A.M. (0-4)**:
   - Height = 12.62 million kWh
   - Area = \( 12.62 \times 2 = 25.24 \) million kWh

2. **4 A.M. to 8 A.M. (4-8)**:
   - Height = 16.62 million kWh
   - Area = \( 16.62 \times 2 = 33.24 \) million kWh

3. **8 A.M. to Noon (8-12)**:
   - Height = 23.29 million kWh
   - Area = \( 23.29 \times 2 = 46.58 \) million kWh

4. **Noon to 4 P.M. (12-16)**:
   - Height = 29.15 million kWh
   - Area = \( 29.15 \times 2 = 58.30 \) million kWh

5. **4 P.M. to 8 P.M. (16-20)**:
   - Height = 20.49 million kWh
   - Area = \( 20.49 \times 2 = 40.98 \) million kWh

6. **8 P.M. to Midnight (20-24)**:
   - Height = 13.48 million kWh
   - Area = \( 13.48 \times 2 = 26.96 \) million kWh

### Total Area (Right Endpoint)
Now sum these areas:

\[
\text{Total Area} = 25.24 + 33.24 + 46.58 + 58.30 + 40.98 + 26.96 = 231.60 \text{ million kWh}
\]

### Average of the Results
Finally, we find the average of the left and right endpoint estimates:

\[
\text{Average} = \frac{226.62 + 231.60}{2} = \frac{458.22}{2} = 229.11 \text{ million kWh}
\]

### Summary
- **Left Endpoint Estimate**: 226.62 million kWh
- **Right Endpoint Estimate**: 231.60 million kWh
- **Average Estimate**: 229.11 million kWh

If you have any questions or need further clarification, feel free to ask! Here are some related questions you might consider:

1. How would the total usage change if we used different width intervals?
2. Can we analyze the data using more sophisticated numerical methods?
3. How does electricity usage vary during different times of the day?
4. What are the implications of this usage for energy policy?
5. How can this data be visualized for better understanding?

**Tip**: For more accurate estimates, consider using more intervals or applying numerical integration techniques like trapezoidal or Simpson's rule.

The graph shows a​ city's daily rate of use of electricity​ (in millions of​ kilowatts). Estimate the total daily usage of electricity by summing the area of rectangles using the left and right endpoints, and then give the average of those results.

Explore how to estimate a city's daily electricity usage by summing rectangle areas with left and right endpoints. This calculation yields a left estimate of 226.62 million kWh, a right estimate of 231.60 million kWh, and an average of 229.11 million kWh. Ideal for advanced high school students.

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