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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.

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00:03

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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All right, we've got a question here. We were told that there is a Lake Lanier in Georgia as a reservoir created by boo for damn buffer. Damn! All right. We have a table that shows the rate of inflow of water and cubic feet per second as measured every morning at 7:30 a.m. And we want to use the midpoint rule to estimate the amount of water that flowed into Lake Linear linear from July 18th, 2013 to July 26th. All right, so to do this, what we're gonna do, Yes, we're going to take how were Well, first, we'll write out an integral from the 18th. Yeah, to the 26th for the function of X. Relax. Alright. Here we want to find our change in X, which would be our B with just 26 minus a, which is 18 over our terms. And and if we divide the interval into four sub intervals, but it's for yeah, my stylist is acting up. You? Mhm. We waited four here. Well, and which is our sub intervals? We'll have 26 minutes. 18, 88 divided by four institute. All right, so then we would write out our intervals with the change in extreme too. So we'll be 18 to 18 to 2022 22 22 24 then 24 to 26 if we use the midpoint rule that we're gonna choose the value in between arrange mental, which is, like we said our ranges from 18 to 20 so we would choose 19. Then we have another interval from 20 to 22. Shall we choose 21 22 to 24 which would be the midpoint would be 23 24 26 with the midpoint of 25? Yes. Then we can write this out by saying the change in X multiplied bye function the effects of I'm equipment. All right, that can be written as to in the function of X. So by midpoint is like I said, the mid point for each intervals added together. So what's the value of July 19th? 6401? Well, suppose the value at 21. 4249. What's what's the value? At 23. 38 21 the value at 25 which is 20 to 6 to 8. Mhm. All right, let's go and plug that into our handy dandy calculator. 2 3000. She's been 34,000, 198 and the units here are feet cubed was second. Okay, because that's where units here are. Yeah, but we're asked to calculate this in the units. Uh, the water that flowed likely in your were asked to solve this for the total amount of water that is in flowed from July 18th to July 26. So we would then take that value that we have. 36,100 98 be cubed and multiplied by the total seconds. Alright. So from July 18th to July 26 we know that's eight days, and then we would have eight days multiplied by 24 hours. We're trying to convert into seconds. Then we know one hour is the same thing as 3600 seconds. So we have eight times 24 times 36 3600, which is equal to So it's 91200 Yes. We had 34,198 ft cubed for a second. So you would be 3498 feet huge multiplied. 697,200 seconds. And then your seconds would cancel out. So you'd be left with a total amount of 2000 ST. You have to point three 64 times 10 to the 10th. Pete. Hugh, Water. I just want to make sure you got that correctly. So you've got 34,198 ft cube per second, and then we wanna convert that to days, and we would have 24 hours multiplied by 36 seconds. And then we multiply that by or eight days. Yeah. That will be your final answer there. All right, well, I hope that clarifies the question. And thank you so much for watching

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