To find lower and upper estimates for the total amount of oil that leaked out, you can use the method of Riemann sums. Given the rate at two-hour intervals, we can approximate the total amount of oil leaked by using the left endpoint and right endpoint approximations.
Let's assume the time intervals are 0-2 hours, 2-4 hours, 4-6 hours, and so on. Here's a simple example using the left-hand and right-hand endpoints:
If the rate was given as:
- At 0-2 hours: r(0) = 100 liters per hour
- At 2-4 hours: r(2) = 80 liters per hour
- At 4-6 hours: r(4) = 60 liters per hour
- And so on...
You can calculate approximate lower and upper estimates for the total amount of oil leaked as follows:
1. Lower Estimate:
- Using the left endpoint approximation, you sum the product of the smallest rate and the time interval for each two-hour period. For example, for the time interval 0-2 hours, you’d use r(0) (i.e., 100 liters per hour) as the approximate rate for that entire period.
- So, for the first two-hour period: 100 liters per hour * 2 hours = 200 liters
- For the second two-hour period: 80 liters per hour * 2 hours = 160 liters
- And so on...
2. Upper Estimate:
- Using the right endpoint approximation, you sum the product of the largest rate and the time interval for each two-hour period. For the time interval 0-2 hours, you’d use r(2) (i.e., 80 liters per hour) as the approximate rate for that entire period.
- So, for the first two-hour period: 100 liters per hour * 2 hours = 200 liters
- For the second two-hour period: 80 liters per hour * 2 hours = 160 liters
- And so on...
By adding up these approximations for each interval, you can find both the lower and upper estimates for the total amount of oil leaked out.
If the exact values of the rates at each two-hour interval are provided, you can use those actual rates to obtain more accurate estimations.
If you have specific rate values for each time interval, please provide them, and I can help you calculate the lower and upper estimates for the total amount of oil leaked out.