The average value of a function
For example, the average value of
What the Average Value Means
Multiply both sides of the formula by
The left side is the area of a rectangle with width
Picture the region as water in a tank with a wavy surface. Let the water settle and it becomes flat. The height of that flat surface is the average value. The parts of the graph above the line fill the gaps below it exactly.
Where the Formula Comes From
For a list of numbers, the average is the sum divided by the count. To average a function, sample it. Split
Since
The sum is a Riemann sum. As
How to Find the Average Value of a Function
- Identify
and the interval . - Integrate: evaluate
with the fundamental theorem of calculus. - Divide by the length
. - Sanity-check: the answer should lie between the smallest and largest values of
on the interval.
Example 1: A Polynomial
Find the average value of
The interval has length
Check:
Example 2: Sine Over Half a Period
Find the average value of
The arch reaches 1, but it spends much of its time lower, so the average is about 0.64. Over a full period
Example 3: The Function 1/x
Find the average value of
Example 4: Using U-Substitution
Find the average value of
Use u-substitution with
Example 5: Average Temperature
The temperature
The interval is
The day peaks at 64°F at noon, but the average over the 12 hours is about 58.9°F.
Example 6: Average Velocity
A particle moves with velocity
The integral of velocity is the displacement (the net change), so the average value of velocity is displacement divided by time, the same “average velocity” you met in algebra.
Average Value Can Be Zero or Negative
The integral counts area below the x-axis as negative, so the average value can be negative. For
That makes sense: the graph spends more of the interval below the axis than above.
Mean Value Theorem for Integrals
The mean value theorem for integrals says a continuous function actually reaches its average value.
Mean value theorem for integrals. If
is continuous on , there is at least one number in such that
or equivalently
The reason is that a continuous function on a closed interval has a minimum
It’s also the mean value theorem in disguise. Let
To find c: compute the average value, set
Example 7. Find the value of
The average value is 3, so
Example 8. Find
From Example 1,
So
There can be more than one
Average Value vs Average Rate of Change
These two are easy to mix up:
| Formula | Measures | |
|---|---|---|
| Average value of | the average height of the graph | |
| Average rate of change of | the average slope of the graph |
They’re connected by the fundamental theorem:
Common Mistakes
- Forgetting to divide by
. The integral alone is an area, not an average. - Dividing by
instead of . On , the length is 3, not 4. - Using
. Averaging the endpoint values only works for linear functions. For on it gives 4.5, not 3. - Confusing average value with average rate of change. One averages
, the other averages . - Keeping a
outside the interval in the mean value theorem for integrals.
For more on both topics, see Sections 5.1 and 5.3 of the free textbook OpenStax Calculus Volume 1.
Practice Problems
Try each one before opening the answer. The practice set has 12 more with instant checking.
Problem 1. Find the average value of
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Answer:
Problem 2. Find the average value of
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Answer:
Problem 3. Find the average value of
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Answer:
Problem 4. Find the average value of
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Answer:
Problem 5. Find the value of
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Answer:
Problem 6. Find the average value of
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With
Answer:
The average value of a function closes the Integrals chapter of Calculus 1. Calculus 2 builds on it with more integration techniques and applications such as area between curves and volume.