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## Finding the average value of a function on an interval

# Average value over a closed interval

AP.CALC:

CHA‑4 (EU)

, CHA‑4.B (LO)

, CHA‑4.B.1 (EK)

## Video transcript

what I want to do in this video is think
about the idea of an average value of a function over some closed interval so
what do I mean by that and how could we think about what average value of a
function even means so let's say that's my y-axis
and let's say that this is my this right over here is my x-axis and let me draw a
function here so let's say the function looks something like that that's the
graph of y is equal to Y is equal to f of X and now let's think about a closed
interval so we're going to think about the closed interval between a and B
including a and B that's what makes it closed or including our endpoints we're
going to think about this interval right over here so between x is equal to a and
X is equal to B what is the average value of this function one way to think
about it is what is the average height of this function so how could we what
would that mean well one way to think about it it would be some height so that
if we multiply it times the width of this interval we'll get the area under
the curve so another way to think about it is the area under the curve right
over here the area let me do this in a different color so the area under this
curve right over here I'll shaded in yellow we already know that we can
express this as the definite integral from A to B of f of X DX the average
value of our function over this closed interval a B let me write that over over
the closed interval between a and B including a and B we could think about
it as some height some height let me do this in a new color some value of our
function some height let me think about maybe some height right over here so
that if we multiply this height times this width we're going to get the area
of a rectangle the rectangle is would be the area of this rectangle right over
here and that rectangle is going to have the same area as the area under the
curve which is a reasonable way if you kind of remember how when you even think
about finding the area or one way to find think about the area of
a trapezoid you can multiply if you have a trapezoid if you have a trapezoid like
this you have a trapezoid like this this is you can kind of you kind of turn 90
degrees but your multiply the height times the average width of the trapezoid
and then that will give you its area so this would be the average width which in
a which in a trapezoid like this would just be halfway between this function is
not linear so it's not necessarily going to be halfway in between but it's that
same idea so how could we use this idea where this is right over here this
height this height right over here we could call this we could call this the
functions average the functions average how could we use all of this to come
back to with a formula for the the the average of a function over this closed
interval well let's just Express in math what we've already said we already said
that this function average should be some height so let's say the function
average so if I that's a height and if I multiply it times the width of this
interval so this width right over here this width right over here is just going
to be the larger value minus the smaller value so that's going to be B minus a so
the average value of the function times the width of the interval should give us
an area that is equivalent to the area under the curve so it should be equal to
the definite integral from A to B of f of X DX and so if we just if we knew all
of this other stuff we could solve for the functions average the functions
average we divide both sides by B minus a the functions average is going to be
equal to just dividing both sides by B minus a you're going to get 1 over B
minus a times the definite integral the definite integral from A to B of f of X
DX or another way to think about it you're going to figure out what the area
under the curve is over that interval you're going to divide that by the width
and then you're going to have the functions average you want me to think
about you're going to have the average height and once again I'd like to remind
you that because this is you shouldn't just sit there and try to memorize this
thing and just just get a conceptual understanding of what this is
really just trying to say area under the curve divided by the width well that's
just going to give you the average height that's going to give you the
average of the function in the next video we'll actually apply this formula
to see that it's actually straightforward to calculate if you if
you can figure out the definite integral

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