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. (Use 6 equal subintervals.)t 0 1 2 3 4 5 6r(t) 2 8 20 32 46 52 60(a) Give upper and lower estimates for the total quantity Q(6) of erupted materials after 6 seconds.Q(6) = tonnes (lower estimate)Q(6) = tonnes (upper estimate)(b) Use the Midpoint Rule to estimate Q(6). (Use 3 equal subintervals.)Q(6) = tonnes
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. (Use 6 equal subintervals.)t 0 1 2 3 4 5 6r(t) 2 8 20 32 46 52 60(a) Give upper and lower estimates for the total quantity Q(6) of erupted materials after 6 seconds.Q(6) = tonnes (lower estimate)Q(6) = tonnes (upper estimate)(b) Use the Midpoint Rule to estimate Q(6). (Use 3 equal subintervals.)Q(6) = tonnes
Solution
(a) To estimate the total quantity Q(6) of erupted materials after 6 seconds, we can use the concept of Riemann sums to approximate the area under the curve.
For the lower estimate, we use the left endpoints of each subinterval. The width of each subinterval is 1 second. So, the lower estimate is:
Q(6) = r(0) + r(1) + r(2) + r(3) + r(4) + r(5) = 2 + 8 + 20 + 32 + 46 + 52 = 160 tonnes
For the upper estimate, we use the right endpoints of each subinterval. So, the upper estimate is:
Q(6) = r(1) + r(2) + r(3) + r(4) + r(5) + r(6) = 8 + 20 + 32 + 46 + 52 + 60 = 218 tonnes
(b) To use the Midpoint Rule, we need to divide the interval [0,6] into 3 equal subintervals: [0,2], [2,4], and [4,6]. The midpoints of these subintervals are 1, 3, and 5. So, the Midpoint Rule gives us:
Q(6) = 2*(r(1) + r(3) + r(5)) = 2*(8 + 32 + 52) = 184 tonnes
So, the lower estimate for Q(6) is 160 tonnes, the upper estimate is 218 tonnes, and the Midpoint Rule estimate is 184 tonnes.
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