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# Calculating average value of function over interval

AP.CALC:
CHA‑4 (EU)
,
CHA‑4.B (LO)
,
CHA‑4.B.1 (EK)

## Video transcript

let's say that we have the function f of X is equal to x squared plus 1 and what we want to do is we want to figure out the average value of our function our function f on the interval on the closed interval between 0 and let's say between 0 and 3 and I encourage you to pause this video and especially if you've seen the other videos on introducing the idea of the average value of a function figure out what this is what is the average value of our function f over this interval so I'm assuming you've had a go at it let's just visualize what's going on and then we can we can actually find the average so that's my y-axis this is my x-axis now over the interval between 0 and 3 so let's say that this is 0 this is 1 2 3 it's a closed interval when when x is 0 f of 0 is going to be 1 so we're going to be we're going to be right over here F of 1 is 2 so it's going to be so it's 1 2 3 actually let me make my scale a little bit smaller on that have to go all the way up to not up to 10 so this is going to be 10 this is going to be 5 and then 1 2 3 actually let me the hardest part is making this even so let's see this is going to be in the middle pretty good and then C in the middle and then we have that not good enough all right so we're going to be we're going to be there we're going to be there I have obviously different scales for x and y-axes 2 squared plus 1 is 5 3 squared plus 1 is 10 3 squared plus 1 is 10 so it's going to look something like this this is what our function is going to look like so that's the graph of y is equal to f of X and we care about the average value on the interval closed interval between 0 and 3 between the 0 and three so one way to think about you can apply the formula but it's very important to think about what does that formula actually mean and once again you shouldn't memorize this formula because it actually kind of falls out of what it actually means so the average of our function is going to be it's going to be equal to the definite integral over this interval so essentially the area under this curve so it's going to be the definite integral from 0 to 3 of f of X which is x squared plus 1 DX and we're going to take this area we're going to take this area right over here and we're going to divide it by the width of our interval to essentially come up with the average height or the average value of our function so we're going to divide it by B minus a or 3-0 which is just going to be 3 and so now we just have to evaluate this so this is going to be equal to 1/3 times C the antiderivative of x squared is X to the third over 3 antiderivative of 1 is X and we're going to evaluate it from 0 to 3 and so this is going to be equal to 1/3 times when we evaluate it at 3 let me use another color here when we evaluated at 3 it's going to be 3 to the third divided by 3 well that's just going to be 27 divided by 3 that's 9 plus 3 and then when we evaluate at 0 minus 0 minus 0 so it's just minus minus when you evaluate at 0 it's just going to be 0 and so we are left with I want to make the brackets that same color this is going to be 1/3 times 12 1/3 times 12 which is equal to 4 which is equal to 4 so this is the average value of our function the average value of our function over this interval over this interval is equal the average value of our function is equal to 4 and notice our function actually hits that value at some point in the interval at some point in the interval something lower then two but greater than one we could maybe call that see it looks like our function hits that value and this is actually this comes this is it this is actually a generally true thing this is a mean value theorem for integrals that and we'll go into more depth there but you can see that this kind of does look like it's average value that if you imagine the box if you multiply this height this average value times this with you would have this area right over here and this area right over here is the same this area that I'm highlighting in yellow right over here is the same as the area under the curve because we have the average height times the width is the same thing as the area under the curve so anyway hopefully you found that interesting
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