Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into

Question

Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for r(t) are tonnes (metric tons) per second.

 

t     0 1    2   3  4   5   6

r(t) 2 12 20 36 44 52 60

 

(a) Give upper and lower estimates for the total quantity Q(6) of erupted materials after 6 seconds using 6 subintervals.

(b) Use the Midpoint with three subintervals to estimate Q(6).

Answer 1

Assuming that r(t) is increasing for all t in [0,6], a lower estimate is obtained by using left endpoints and an upper estimate is achieved by using right endpoints.  

 

Lower estimate:  r(0)(1)+r(1)(1)+r(2)(1)+r(3)(1)+r(4)(1)+r(5)(1)

 

                       = 2+12+20+36+44+52 = 166 metric tons

 

Upper estimate:  r(1)(1)+r(2)(1)+r(3)(1)+r(4)(1)+r(5)(1)+r(6)(1)

 

                       = 12+20+36+44+52+60 = 224 metric tons

 

Midpoint estimate:  r(1)(2)+r(3)(2)+r(5)(2)

 

                          = 24+72+104 = 200 metric tons