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

Question

t (hr)r(t) (L/h)
0	10.9
4	10.3
8	9.7
12	9.2
16	8.7
20	8.2

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

(a) lower estimate =  liters

(b) upper estimate =  liters

Answer 1

Use a Riemann sum approach to find areas under the r(t) curve. The left endpoint method will give the upper estimate, and the right endpoint method will give the lower estimate.

Upper (left endpoint): 4*10.9 + 4*10.3 + 4*9.7 + 4*9.2 +4*8.7 = 195.2

Lower (right endpoint): 4*10.3 + 4*9.7 + 4*9.2 + 4*8.7 + 4*8.2 = 184.4

You're simply drawing rectangles of width 4 and length determined by r(t). Then just add up these areas.