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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

Lake Lanier in Georgia, USA, is a reservoir created by Buford Dam on the Chattahoochee River. The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 am by the US Army Corps of Engineers. Use the Midpoint Rule to estimate the amount of water that flowed into Lake Lanier from July 18th, 2013, at 7:30 am to July 26th at 7:30 am.

Day Inflow rate (ft3/s)
July 18 5275
July 19 6401
July 20 2554
July 21 4249
July 22 3016
July 23 3821
July 24 2462
July 25 2628
July 26 3003

To determine

To calculate: The amount of water flowed into the lake from July 18th, 2013 to July 26th, 2013 by using the midpoint rule.

Explanation

Given information:

The day July 18 corresponds to time t=0.

The day July 26 corresponds to time t=8days.

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 subintervals is n, and the midpoint of (xi1,xi) is x¯i.

Calculation:

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

Let us consider the number of subintervals as 4.

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

Δt=804=2day

The expression to find the amount of water flowed into the lake using midpoint rule is shown below.

04r(t)dtΔt(r(t¯1)+r(t¯2)+r(t¯3)+r(t¯4)) (3)

Here, the amount of water flowed into the lake at time t¯1 is r(t¯1), the amount of water flowed into the lake at time t¯2 is r(t¯2), the amount of water flowed into the lake at time t¯3 is r(t¯3), and the amount of water flowed into the lake at time t¯4 is r(t¯4)

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