5.5 Average Value of a Function
It is easy to calculate the average value of finitely many numbers y1 , y2 , ..., yn :
Question: How do we compute the average value of infinitely many values determined by a function
f (x) over and interval [a, b]?
Thought Process: We want to try to find the average value of y = f (x) on a ≤ x ≤ b.
• Divide the interval [a, b] into n equal subintervals, each with length ∆x =
b−a
.
n
• Choose a sample point x∗i in each subinterval [xi−1 , xi ].
• Calculate the average of the numbers f (x∗1 ), f (x∗2 ), ..., f (x∗n ):
• Since ∆x =
b−a
, we can write n =
n
and the average value becomes:
• If we let n increase, we would be computing the average value of a large number of closely spaced
values. The limiting value is
by the definition of a definite integral.
Therefore, we define the
on the interval [a, b] as
1
Example 1. Find the average value of the function on the given interval.
a) f (x) = sin 4x on [−π, π]
b) h(r) =
3
on [1, 6]
(1 + r)2
2
Question: Is there a number c at which the value of a function f is exactly equal to the average value
of the function, that is, f (x) = fave ?
The Mean Value Theorem for Integrals: If f is continuous on [a, b], then there exists a number
c in [a, b] such that
that is,
The Mean Value Theorem for Integrals is a result of the Mean Value Theorem for derivatives and the
Fundamental Theorem of Calculus.
Geometric Interpretation: For positive functions f , there is a number c such that the rectangle with
base [a, b] and height f (c) has the same area as the region under the graph of f from a to b.
Example 2. a) Find the average value of f (x) =
√
x on [0, 4].
b) Find c such that fave = f (c)
c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f .
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Example 3. Find the numbers b such that the average value of f (x) = 2 + 6x − 3x2 on the interval
[0, b] is equal to 3.
Example 4. The velocity v of blood that flows in a blood vessel with radius R and length l at a distance
r from the central axis is
v(r) =
P
(R2 − r2 )
4ηl
where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood. Find
the average velocity (with respect to r) over the interval 0 ≤ r ≤ R. Compare the average velocity with
the maximum velocity.
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