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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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
( ft 3 / 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 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

Explanation

1) Concept:

Use the formula for the amount of water that flowed into Lake Lanier and calculate it by using the midpoint rule.

2) Formula:

The amount of water flowed is t1t2r(t)dt where r(t) is the inflow rate during the time t1 to t2.

3) Calculation:

Let r(t) be the in flow rate which is given in ft3/s.

The amount of water that flowed into Lake Lanier from July 18th, 2013, at 7:30 AM to July 26th, at 7:30 AM, that is, in 8 days is given by

08r(t)dt

Now, find 08r(t)dt by using the midpoint rule,

With n=4, a=0 and b=8 we have the width of interval is

t=b-an

Substitute the values;

t=8-04

t=2

And the midpoints are

t1-=12t0+t1=120+2=122=1

t2-=12t1+t2=122+4=126=3

t3-=12t2+t3=124+6=1210=5<

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