Intuition The one core ideaTo average infinitely many function values, we add them all up with an integral (a "continuous sum") and divide by the length of the interval. So this whole topic is built from just two ideas glued together: summing (which becomes the integral) and dividing by how many (which becomes dividing by the width b − a
This page assumes nothing . On the parent note you will meet a formula that averages a function; the shorthand for "the average value of f f avg name for the answer we are building toward . Before you meet that formula on the parent note , we build every single symbol in it, one at a time, each with a picture and a reason it has to be there.
A function is a machine: you feed it a number x f ( x ) f x f ( x )
Picture it as a curve. Every point on the horizontal axis is an input x height of the curve above that point is the output f ( x )
Intuition Why the topic needs this"Average value" is the average of all those heights . If you can't see f ( x )
Why so-many-values matters A function has one height for every x
Definition Closed interval[ a , b ] [ a , b ] a b a left endpoint (start) and b right endpoint (end). We always take b > a
On the picture: a b between them is the only part we care about.
Intuition Why the topic needs thisYou can only average over a definite stretch. "The average of x 2 x 2 on [ 0 , 3 ] " has one answer. The interval is what makes the question well-posed.
Definition Width of the intervalb − a length of the interval — how far you travel from the start a b b > a positive .
Picture: the length of the base of the shaded region, measured along the horizontal axis.
Intuition Why the topic needs thisWhen you average n n width b − a
a − b b − a Why it tempts: under pressure the order of letters gets scrambled.
Why it's wrong: a width can't be negative. a − b b > a Fix: big minus small. Width = b − a > 0
Now the biggest symbol. Let's build it in pieces.
Intuition Why "area" answers "sum"Imagine slicing the shaded region into thin vertical strips. Each strip is (almost) a rectangle: its height is f ( x ) width is a tiny number we call d x × = f ( x ) d x Add up every strip's area and you have the total area. That "add up all the strips" is what the stretched-S symbol ∫ Sum .
Definition The definite integral, symbol by symbol∫ a b f ( x ) d x
∫ a b f ( x ) height of each strip.d x infinitely thin width of each strip.Together: the total signed area between the curve and the horizontal axis, from a b
"Signed" matters: strips below the axis (where f ( x ) negative area. See §6.
Intuition Why the topic needs thisThe integral is the "add up all the (infinitely many) heights" step. Average value = this total, spread evenly. Without the integral there is no way to sum infinitely many heights — see Definite Integral as Area and Riemann Sums for how the sum is made rigorous.
The parent's derivation passes through this, so let's earn every symbol.
Definition Sigma notation∑ i = 1 ∑ n f ( x i ) f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) ∑ Sum ); i 1 n x i i
Definition Where does each sample pointx i [ a , b ] n Δ x = n b − a i one input to measure the height at — call it x i
the left endpoint of the strip: x i = a + ( i − 1 ) Δ x
the right endpoint : x i = a + i Δ x
the midpoint , or literally any point inside the strip.
f ( x i )
Intuition Why the choice stops mattering in the limitDifferent choices give slightly different staircases (look at the figure: left-endpoint bars vs right-endpoint bars). But as the strips get thinner (n → ∞ all choices converge to the same integral . That is exactly why the notation ∑ f ( x i ) Δ x x i
Δ x n
n Δ x = n b − a Δ
Definition When is this limit guaranteed to exist? (integrability)The staircase sum only settles to a single number if the function is well-behaved . The safe, sufficient condition we use throughout this topic is:
f is == continuous == on [ a , b ] . integrable , so its Riemann sums have a limit and ∫ a b f d x
every function has an average value
Why it tempts: the formula looks like it always works.
Why it's wrong: if f Fix: check f [ a , b ]
Intuition Why the topic needs thisThis is the bridge in the derivation. The finite average n 1 ∑ f ( x i ) b − a 1 ∑ f ( x i ) Δ x ∫
You must handle every case, not just curves that stay above the axis.
Definition Sign of a height
f ( x ) > 0 above the axis → area counts positive .f ( x ) < 0 below the axis → area counts negative .f ( x ) = 0
Intuition Why the topic needs thisThe average value can be negative (if the curve mostly sits below the axis) or exactly zero (if positive and negative parts cancel). Example: f ( x ) = sin x [ 0 , 2 π ] 0
Before we compute that example, we need one more tool — a way to actually evaluate an integral without summing infinitely many strips by hand.
Definition Antiderivative and the evaluation bracketAn antiderivative of f F f F f − cos x sin x − cos x sin x
The Fundamental Theorem of Calculus then says: the total signed area equals the antiderivative measured at b a evaluation bracket :
∫ a b f ( x ) d x = [ F ( x ) ] a b = F ( b ) − F ( a ) . [ ⋯ ] a b
Worked example A zero-average caseUsing F ( x ) = − cos x sin x ∫ 0 2 π sin x d x = [ − cos x ] 0 2 π = ( − cos 2 π ) − ( − cos 0 ) = ( − 1 ) − ( − 1 ) = 0. f avg = 2 π 1 ⋅ 0 = 0
n → ∞ lim ( expression ) n closer and closer to, never needing to reach infinity itself.
Picture: watch the staircase of rectangles from §5. With n = 4 n = 20 n = 1000
Intuition Why the topic needs this"Average of infinitely many" is not something we can compute directly — we compute the average of n take the limit . The limit is what upgrades an approximation into an exact continuous average.
Definition Notation for "average"A bar over a letter, like y ˉ f ˉ f avg
Now every piece of
f avg = b − a 1 ∫ a b f ( x ) d x f avg b − a 1 ∫ a b f ( x ) d x f Area over width — nothing unexplained remains.
The foundations stack up in this order. Read it as "the box below is needed before the box above works":
Function as a height f ( x ) interval [ a , b ] From the interval you get the width b − a
Chopping the interval and summing sampled heights gives the Riemann sum ∑ f ( x i ) Δ x sample-point choice and continuity/integrability to be well-defined.
Taking the limit n → ∞ definite integral ∫ a b f d x antiderivative bracket (§6).
Integral over width gives the average value f avg Mean Value Theorem for Integrals .
Recall Self-test: can you answer each before peeking?
What does f ( x ) The output of the machine f x height of the curve above the point x
What does the interval [ a , b ] Every number from a b b > a
Why must the width be b − a a − b A width is positive; since b > a = b − a > 0
In ∫ a b f ( x ) d x ∫ a b f ( x ) d x ∫ a , b f ( x ) d x
Why does the integral compute a "sum"? Each thin strip has area f ( x ) d x
Where inside each strip does the sample point x i Anywhere in the strip — left endpoint, right endpoint, midpoint, or any point; in the limit the choice doesn't matter.
What condition on f f [ a , b ]
What does ∑ i = 1 n f ( x i ) Δ x n → ∞ The definite integral ∫ a b f ( x ) d x
What is Δ x n Δ x = n b − a n
What does the evaluation bracket [ F ( x ) ] a b F ( b ) − F ( a ) F f
What does lim n → ∞ The single number the expression settles toward as n
What happens to area where f ( x ) < 0 It counts as negative area, which can drag the average below zero or cancel positive parts to zero.
What is the average value of sin x [ 0 , 2 π ] 0
