Average value of a function worksheet

Flatten the function until it is a straight line that shuts in the same area. That height is its average value, and it is the integral divided by the width of the interval.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Average Value of a Function Date Period Find the average value of each function on the given interval. Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$ Find the average value of $f (x) = - 2 x^{2} - 3 x + 6$ on $[2, 4]$ Find the average value of $f (x) = 2 x^{2} - 2 x$ on $[0, 1]$ Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$ Find the average value of $f (x) = - 4 x^{2} - 4 x + 3$ on $[2, 5]$ Find the average value of $f (x) = - x^{2} - 2 x + 3$ on $[3, 4]$ Find the average value of $f (x) = - 3 x^{2} - 2 x + 1$ on $[0, 3]$ Find the average value of $f (x) = - 3 x^{2} + 3 x + 6$ on $[3, 4]$ Find the average value of $f (x) = x^{2} + 5 x + 3$ on $[3, 7]$ Find the average value of $f (x) = 3 x^{2} + 3 x - 3$ on $[0, 4]$ Find the average value of $f (x) = - 2 x^{2} - 5 x + 6$ on $[0, 3]$ Find the average value of $f (x) = x^{2} + 3 x - 2$ on $[0, 2]$ Find the average value of $f (x) = 2 x^{2} - 2 x - 3$ on $[3, 7]$ Find the average value of $f (x) = - x^{2} - x + 1$ on $[0, 3]$ Find the average value of $f (x) = x^{2} - 3 x + 2$ on $[1, 4]$ Find the average value of $f (x) = - x^{2} + 5 x + 6$ on $[3, 5]$ Find the average value of $f (x) = - 2 x^{2} + 5 x - 2$ on $[0, 4]$ Find the average value of $f (x) = - 2 x^{2} - x + 4$ on $[3, 7]$ Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$ Find the average value of $f (x) = - 4 x^{2} - 2 x - 2$ on $[2, 3]$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Average Value of a Function

Date Period

Find the average value of each function on the given interval.

Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$
Find the average value of $f (x) = - 2 x^{2} - 3 x + 6$ on $[2, 4]$
Find the average value of $f (x) = 2 x^{2} - 2 x$ on $[0, 1]$
Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$
Find the average value of $f (x) = - 4 x^{2} - 4 x + 3$ on $[2, 5]$
Find the average value of $f (x) = - x^{2} - 2 x + 3$ on $[3, 4]$
Find the average value of $f (x) = - 3 x^{2} - 2 x + 1$ on $[0, 3]$
Find the average value of $f (x) = - 3 x^{2} + 3 x + 6$ on $[3, 4]$
Find the average value of $f (x) = x^{2} + 5 x + 3$ on $[3, 7]$
Find the average value of $f (x) = 3 x^{2} + 3 x - 3$ on $[0, 4]$
Find the average value of $f (x) = - 2 x^{2} - 5 x + 6$ on $[0, 3]$
Find the average value of $f (x) = x^{2} + 3 x - 2$ on $[0, 2]$
Find the average value of $f (x) = 2 x^{2} - 2 x - 3$ on $[3, 7]$
Find the average value of $f (x) = - x^{2} - x + 1$ on $[0, 3]$
Find the average value of $f (x) = x^{2} - 3 x + 2$ on $[1, 4]$
Find the average value of $f (x) = - x^{2} + 5 x + 6$ on $[3, 5]$
Find the average value of $f (x) = - 2 x^{2} + 5 x - 2$ on $[0, 4]$
Find the average value of $f (x) = - 2 x^{2} - x + 4$ on $[3, 7]$
Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$
Find the average value of $f (x) = - 4 x^{2} - 2 x - 2$ on $[2, 3]$



Sigma Prep · sigmaprep.io/worksheets Name Average Value of a FunctionAnswer keyVersion 1 Date Period Find the average value of each function on the given interval. Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$ $- \frac{13}{2}$ Find the average value of $f (x) = - 2 x^{2} - 3 x + 6$ on $[2, 4]$ $- \frac{65}{3}$ Find the average value of $f (x) = 2 x^{2} - 2 x$ on $[0, 1]$ $- \frac{1}{3}$ Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$ $\frac{2}{3}$ Find the average value of $f (x) = - 4 x^{2} - 4 x + 3$ on $[2, 5]$ $- 63$ Find the average value of $f (x) = - x^{2} - 2 x + 3$ on $[3, 4]$ $- \frac{49}{3}$ Find the average value of $f (x) = - 3 x^{2} - 2 x + 1$ on $[0, 3]$ $- 11$ Find the average value of $f (x) = - 3 x^{2} + 3 x + 6$ on $[3, 4]$ $- \frac{41}{2}$ Find the average value of $f (x) = x^{2} + 5 x + 3$ on $[3, 7]$ $\frac{163}{3}$ Find the average value of $f (x) = 3 x^{2} + 3 x - 3$ on $[0, 4]$ $19$ Find the average value of $f (x) = - 2 x^{2} - 5 x + 6$ on $[0, 3]$ $- \frac{15}{2}$ Find the average value of $f (x) = x^{2} + 3 x - 2$ on $[0, 2]$ $\frac{7}{3}$ Find the average value of $f (x) = 2 x^{2} - 2 x - 3$ on $[3, 7]$ $\frac{119}{3}$ Find the average value of $f (x) = - x^{2} - x + 1$ on $[0, 3]$ $- \frac{7}{2}$ Find the average value of $f (x) = x^{2} - 3 x + 2$ on $[1, 4]$ $\frac{3}{2}$ Find the average value of $f (x) = - x^{2} + 5 x + 6$ on $[3, 5]$ $\frac{29}{3}$ Find the average value of $f (x) = - 2 x^{2} + 5 x - 2$ on $[0, 4]$ $- \frac{8}{3}$ Find the average value of $f (x) = - 2 x^{2} - x + 4$ on $[3, 7]$ $- \frac{161}{3}$ Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$ $18$ Find the average value of $f (x) = - 4 x^{2} - 2 x - 2$ on $[2, 3]$ $- \frac{97}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Average Value of a FunctionAnswer keyVersion 1

Date Period

Find the average value of each function on the given interval.

Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$ $- \frac{13}{2}$
Find the average value of $f (x) = - 2 x^{2} - 3 x + 6$ on $[2, 4]$ $- \frac{65}{3}$
Find the average value of $f (x) = 2 x^{2} - 2 x$ on $[0, 1]$ $- \frac{1}{3}$
Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$ $\frac{2}{3}$
Find the average value of $f (x) = - 4 x^{2} - 4 x + 3$ on $[2, 5]$ $- 63$
Find the average value of $f (x) = - x^{2} - 2 x + 3$ on $[3, 4]$ $- \frac{49}{3}$
Find the average value of $f (x) = - 3 x^{2} - 2 x + 1$ on $[0, 3]$ $- 11$
Find the average value of $f (x) = - 3 x^{2} + 3 x + 6$ on $[3, 4]$ $- \frac{41}{2}$
Find the average value of $f (x) = x^{2} + 5 x + 3$ on $[3, 7]$ $\frac{163}{3}$
Find the average value of $f (x) = 3 x^{2} + 3 x - 3$ on $[0, 4]$ $19$
Find the average value of $f (x) = - 2 x^{2} - 5 x + 6$ on $[0, 3]$ $- \frac{15}{2}$
Find the average value of $f (x) = x^{2} + 3 x - 2$ on $[0, 2]$ $\frac{7}{3}$
Find the average value of $f (x) = 2 x^{2} - 2 x - 3$ on $[3, 7]$ $\frac{119}{3}$
Find the average value of $f (x) = - x^{2} - x + 1$ on $[0, 3]$ $- \frac{7}{2}$
Find the average value of $f (x) = x^{2} - 3 x + 2$ on $[1, 4]$ $\frac{3}{2}$
Find the average value of $f (x) = - x^{2} + 5 x + 6$ on $[3, 5]$ $\frac{29}{3}$
Find the average value of $f (x) = - 2 x^{2} + 5 x - 2$ on $[0, 4]$ $- \frac{8}{3}$
Find the average value of $f (x) = - 2 x^{2} - x + 4$ on $[3, 7]$ $- \frac{161}{3}$
Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$ $18$
Find the average value of $f (x) = - 4 x^{2} - 2 x - 2$ on $[2, 3]$ $- \frac{97}{3}$

How to do these

Find the average value of $f (x) = x^{2}$ on $[0, 3]$

The integral of x squared from 0 to 3 is 9.
The interval is 3 wide.
So the average value is 9 over 3, which is 3.

The answer is $3$ .

Where students go wrong

Forgetting to divide by the width. The integral is the area, not the height, and it only becomes an average once it is spread across the interval it came from.

Also called mean value theorem for integrals, average value or mean value of a function.

Questions about these worksheets

Yes. Every average value of a function sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the average value of a function questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Forgetting to divide by the width. The integral is the area, not the height, and it only becomes an average once it is spread across the interval it came from.

Average value of a function worksheet

Select your difficulty

Average Value of a Function

Average Value of a FunctionAnswer keyVersion 1

How to do these

Where students go wrong

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