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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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.

Chapter 5.4, Problem 66E, Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed

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

(b) Use the Midpoint Rule to estimate Q(6).

(a)

To determine

To find: The total quantity of upper and lower estimates of erupted materials after six seconds duration.

Explanation

Given information:

The region lies between t=0 and t=6sec.

Calculation:

The expression to find the upper estimate for total quantity of erupted materials (Rn) is shown below:

Rn=r(t1)Δt+r(t2)Δt+...+r(tn)Δt (1)

Here, the rate of volcanic eruption at time t1 is r(t1), the time interval is Δt, the rate of volcanic eruption at time t2 is r(t2), and the rate of volcanic eruption at time tn is r(tn)

Substitute 6 for n in Equation (1).

R6=r(t1)Δt+r(t2)Δt+r(t3)Δt+r(t4)Δt+r(t5)Δt+r(t6)Δt (2)

Substitute 10tonnes/sec for r(t1), 1 sec for Δt, 24tonnes/sec for r(t2), 36tonnes/sec for r(t3), 46tonnes/sec for r(t4), 54tonnes/sec for r(t5), and 64tonnes/sec 64 for r(t6) in Equation (2).

R6=(10×1)+(24×1)+(36×1)+(46×1)+(54×1)+(64×1)=10+24+36+46+54+64=234tonnes

Therefore, the upper estimate for total quantity of erupted materials is 234tonnes_

(b)

To determine

To find: The total quantity of erupted materials using Midpoint Rule.

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