Assignment due at midnight, please help!!

Question

Liquid leaked from a damaged tank at a rate of  liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table.

t (hr)
0	9.6
2	9.3
4	8.9
6	8.6
8	8.3
10	8

Find lower and upper estimates for the total amount of liquid that leaked out.

Answer 1

I would recommend taking the left and right endpoint approximations for the value of the integral. To do this, for each region, you calculate the area of the rectangle first utilizing the value of the derivative at the left endpoint, so for example, the first rectangle would be between t=0 and t=2 and would be approximated using the left endpoint value of 9.6 liters/hour, so the area would be 2*9.6=19.2 liters. You then repeat this process for the remaining intervals, utilizing the value of the rate at the left endpoint of the interval. Since the rate is decreasing over time, you know that the average rate across the interval will be less than the rate at the left endpoint, so this approximation will be the upper bound for the area in each interval. Then, summing these areas, you can find the upper estimate for the total liquid that leaked out.

To find the lower estimate, you would follow the same process, except using the value of the rate at the right endpoint since this will be less than the average rate across the interval.