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In Example 3.8.1 we modeled the world population in the second half of the 20th century by the equation$ P(t) = 2560e^{0.017185t} $. Use this equation to estimate the average world population during this time period.

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Amrita B.

03:18

Bobby B.

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Willis J.

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 5

Average Value of a Function

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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Hi there. So for this problem we have model of the world population with an equation in function of the time. E. And now we are asked to use this equation to estimate the average world population during the period of time that is Half of the 12 of the 20th century. So we know that the half of the 20th century um they can halt is fraud 19 52 The 2000s. So we can see him here. Yeah A difference in crying is 50 years. So we are going to set ah 1950 is equal to equal to zero and the filter and far 50 years later. So with this in mind we can use the equation for the average. The honourable judge equation is yes. One over the ulker limits minus the lower limit. The integral of this function from A to P. Of it's something. No we know that. A wait. Well in this case be it's 15 and a zero a zero 15. And we need to boot this equation. So now we're gonna take out of the out of indices. We are integrated in the variable. So we're gonna take out of the integral everything that is constant. We can see that this number so we can take it out of the angel role. So we are going to do not one. Um Here. Yeah. Yeah. Right. No. To solve this integral more easily more easy. Uh easily we can said We can change the variable so that we make U equal to 0.0 17 18 fine theme and then we we terrified expression its parents. Now we can from this last one we can so for L. T. Or if the time differential so that we are going to obtain this. Um With this done we are going to put this into the internal we're not going to change the interval. We are going to do this and paint and later we're gonna get back to the empty variable. So when we do that weekend uh huh. We get these in the denominator and in the wrong we're gonna they've had without limit that we apprehended and we know that the into law of the an exponential function is just the exponential function. And we also so it is the international function but in this case we need to um able weight is incredible with its incredible. So we just and back to the three the variable. So using this again we just can't simply changed. Got in here so it is 0.0 And this is from 0 to 15. Um to evaluate it which simply when we evaluate an incurable we first evaluate um the upper limit minus the function in the lower limit. So are you going to get in the upper limit we obtain minus the exponential of zero because We are not decline a number by zero And we know that the exponential at zero is just one. So we just think you could all of this into the calculator to obtain a number. So we will have something like this and times it's so it is the best inimitable putting all these values into the calculator. We obtain 4015 up and fit 0.4. And this is the average or population during this period of time. Remember that these expression is in millions. So yes. Okay. The world population during that period of time. Thank you.

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