Section 6.1

Average Function Value

To find the average of a list of numbers, you sum them and divide by the count. To find the average of a continuous function, you integrate (sum) and divide by the width.

The Leveling Principle

Think of the function as water in a tank. If the water settles, what is the height?

The Liquid Leveler

Average Value = Leveling the waves

Notice: The total amount of blue water (Area) never changes. The height just averages out.

Worked Example: Temperature

Average Temp

Temp is over 24 hours ( to ). Find average.

1. Width

2. Integrate

. (Sine part sums to 0 over full cycle).

3. Divide

degrees.

Level Up Examples

Example A: Parabola Average

Find avg value of on .

1. Setup:

2. Integrate:

3. Result:

. (Also,

, Mean Value Theorem).

Example B: Average Speed

Velocity on .