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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

Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 am. D is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period.

Chapter 5.4, Problem 72E, Shown is the graph of traffic on an Internet service providers T1 data line from midnight to 8:00

To determine

To calculate: The total amount of data transmitted during 8 hours’ time period.

Explanation

Given information:

Duration of data transmission is t=0 to t=8hours.

The general expression for midpoint rule is shown below:

abf(x)dxi=1nf(x¯i)Δx=Δx(f(x¯1)+....+f(x¯n))

Apply the conditions for midpoint rule as follows:

Δx=ban (1)

x¯i=12(xi1+xi) (2)

Here, width of the subinterval is Δx, the upper limit is b, the lower limit is a, the number of subinterval is n, and the midpoint of (xi1,xi) is x¯i.

Calculation:

Find the width of the subinterval (Δt) using Equation (1).

Consider the number of subintervals as 4.

Substitute 4 for n, 8 hours for b and 0 for a in Equation (1).

Δt=804=2hours

The expression to find the total amount of data transmitted [D(8)D(0)] during the time t=0 to t=8hours using midpoint rule as shown below.

[D(8)D(0)]=08D(t)dtΔt(D(t¯1)+D(t¯2)+D(t¯3)+D(t¯4)) (3)

Here, the amount of data transmitted at time t¯1 is D(

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