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# Calculating average value of function over interval

Here we find the average value of x^2+1 on the interval between 0 and 3.

Video transcript

Let's say that we have the function, F of
X is equal to X squared plus one. And what we want to do is we wanna figure
out the average value of our function our function F on the
interval, on the closed interval between zero and, let's say
between zero and three. And I encourage you to pause this video,
and especially if you see in 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 zero and
three, so lets say this is zero, this is one, two,
three. It's a close interval. When, when X is zero, F of zero is going
to be one. So, we're going to be, we're going to be
right over here. F of one is two, so it's gonna be, so this
is one, two, three. Actually, let me make my scale a little
bit smaller on that. I have to go all the way up to nine, up to
ten. So, this is gonna be ten. This is gonna be five and then one, two,
three. Actually let me, it's hardest part. It's making this even. So let's see. This gonna be in the middle. Pretty good. And then let's see, in the middle and then
we have that. Oh good enough. All right. So we're gonna be, we're gonna be there. We're gonna be there. I have obviously different scales for X
and Y axis. Two squared plus one is five. Three squared plus one is ten. Three squared plus one is ten. So it's going to look something like this. This is what our function is going to look
like. So that is the graph of Y is equal to F of
X. And we care about the average value of the
interval, close interval, between zero and three,
between zero and three. So one way to think about it, 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 kind of actually 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 zero to three of F of X, which is X squared plus
one DX. And we're gonna take this area, we're
gonna take this area right over here and we're gonna
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 gonna divide it by B minus A, or three minus zero, which is just going to
be three. And so now we just have to evaluate this. So this is going to be equal to one third
times, see, the antiderivative of X squared is X to
the third, or three. Antiderivative of one is X. And we're going to evaluate it from zero
to three. And so this is going to be equal to one
third times, when we evaluate it at three, let me use
another color here. When we evaluate it at three, it's going
to be three to the third divided by three. Well that's just going to be 27 divided by
three is nine plus three. And then when we evaluate at zero minus
zero minus zero. So it's just a minus. Minus when you evaluate zero, it's just
going to be zero. And so, we are left with, I want to make
the brackets that same color. This is going to be one third times 12. One-third times 12 which is equal to four,
which is equal to four. 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 to four. And notice, our function actually hits
that value at some point in the interval. At some point in the interval, something
lower than two but bigger than one, we could maybe call
that C. It looks like our function hits that
value. And this is actually, this comes, this is, this is actually a, 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 width, 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. Cuz we have the average height times the
width is the same thing as the area under the
curve. So anyway, hopefully you've found that
interesting.