Calc home

Average Value of a Function (Calculus AB)

This rundown covers the average value of a function over an interval, a key concept in integral calculus.

Definition

The average value of a function $ f(x) $ on the interval $ [a, b] $ is given by:

$$ f_{avg} = \frac{1}{b-a} \int_a^b f(x) , dx $$
Explanation:

• $ f_{avg} $ represents the average y-value of the function $ f(x) $ over the interval $ [a, b] $ .
• $ \int_a^b f(x) , dx $ represents the definite integral of $ f(x) $ from $ a $ to $ b $ , which geometrically represents the area under the curve of $ f(x) $ between $ x = a $ and $ x = b $ .
• $ b - a $ represents the length of the interval.

Essentially, we are dividing the area under the curve by the length of the interval to find the average height (y-value) of the function.

Intuition

Imagine a rectangle with width $ (b - a) $ whose area is equal to the area under the curve of $ f(x) $ from $ a $ to $ b $ . The height of this rectangle is the average value of the function, $ f_{avg} $ .

Formula Derivation

The average value formula comes from the Mean Value Theorem for Integrals. Mean Value Theorem

The Mean Value Theorem for Integrals states that if $ f(x) $ is continuous on $ [a, b] $ , then there exists a value $ c $ in $ [a, b] $ such that:

$$ \int_a^b f(x) , dx = f(c)(b - a) $$
Where $ f(c) $ is the value of the function at some point $ c $ in the interval.

To find the average value, we are essentially finding the constant value of the function, lets call it $ f_{avg} $ , that would give the same area under the curve as the original function over the interval $ [a, b] $ .

So,

$$ \int_a^b f(x) , dx = f_{avg}(b - a) $$
Solving for $ f_{avg} $ , we get:

$$ f_{avg} = \frac{1}{b-a} \int_a^b f(x) , dx $$

Steps to Calculate the Average Value

1. Identify the function: Determine the function $ f(x) $ whose average value you need to find.
2. Identify the interval: Determine the interval $ [a, b] $ over which you want to find the average value.
3. Compute the definite integral: Evaluate the definite integral $ \int_a^b f(x) , dx $ . This might involve using integration techniques like u-substitution.
4. Apply the formula: Plug the result of the integral and the interval endpoints into the average value formula: $ f_{avg} = \frac{1}{b-a} \int_a^b f(x) , dx $ .
5. Simplify: Simplify the expression to find the average value $ f_{avg} $ .

Examples

Example 1:

Find the average value of $ f(x) = x^2 $ on the interval $ [0, 2] $ .

1. Function: $ f(x) = x^2 $
2. Interval: $ [0, 2] $
3. Definite Integral: $$ \int_0^2 x^2 , dx = \left[ \frac{x^3}{3} \right]0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} $$ 4. Average Value: $$ f{avg} = \frac{1}{2-0} \int_0^2 x^2 , dx = \frac{1}{2} \cdot \frac{8}{3} = \frac{4}{3} $$
   Therefore, the average value of $ f(x) = x^2 $ on the interval $ [0, 2] $ is $ \frac{4}{3} $ .

Example 2:

Find the average value of $ f(x) = \sin(x) $ on the interval $ [0, \pi] $ .

1. Function: $ f(x) = \sin(x) $
2. Interval: $ [0, \pi] $
3. Definite Integral: $$ \int_0^\pi \sin(x) , dx = \left[ -\cos(x) \right]0^\pi = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2 $$ 4. Average Value: $$ f{avg} = \frac{1}{\pi - 0} \int_0^\pi \sin(x) , dx = \frac{1}{\pi} \cdot 2 = \frac{2}{\pi} $$
   Therefore, the average value of $ f(x) = \sin(x) $ on the interval $ [0, \pi] $ is $ \frac{2}{\pi} $ .

Common Mistakes

• Forgetting the $ \frac{1}{b-a} $ factor: This is a very common mistake. Make sure to divide the definite integral by the length of the interval.
• Incorrectly evaluating the definite integral: Pay close attention to the limits of integration and the antiderivative.
• Mixing up the average value with the average rate of change: The average value is about the y-values of the function, while the average rate of change is about the slope of the function. Average Rate of Change vs Average Value

Applications

The average value of a function has applications in various fields:

• Physics: Finding the average velocity, average acceleration, etc.
• Engineering: Calculating the average power consumption, average temperature, etc.
• Economics: Determining the average cost, average revenue, etc.

Important Integration Techniques

Remember to review your integration techniques, especially u-substitution, as they will be necessary to evaluate the definite integrals required for calculating the average value of a function.