Overview

How do you take the average of $0.5x^2 + 1$ from $3$ to $8$? Well, we can use something called integral averages!

1. Introduction

An integral average is a method of taking the average of a function across a range by using integrals. It relies on the following formula:

This simply finds the area under the relevant portion of the curve and divides it by the length — to find the height it would be if the shape were a rectangle, which is the average height and thus the average value.

2. Worked Example

Applying the formula to the problem of averaging $0.5x^2 + 1$ from $3$ to $8$:

And there's our answer!

3. Practice Problems

Take the average of the following functions from $-4$ to $4$:

Function	Answer
$8x^3 + 4x + 1$	$1$
$x^2 + 2x + 3$	$25/3$
$9x^3 + x + 7$	$7$
$4x + 1024$	$1024$
$8x^4$	$409.6$
$1834567892345679x + 45678183$	$45678183$
$1$	$1$