Average Value of a Function
Mean height of a curve over an interval.
Key idea
Mean height of a curve over an interval.
In depth
f_avg = (1/(b − a)) ∫ₐᵇ f(x) dx. It is the height of the rectangle with the same area under the curve over the same interval.
Worked example
f(x) = x² on [0, 3]: (1/3)(9) = 3.
Formula
Real-world uses
- Finding the average temperature over a day from a continuous temperature curve
Common mistake
Forgetting to divide by the length of the interval.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Average Value of a Function?
Mean height of a curve over an interval. It is a topic at the advanced level in Calculus.
What is the formula for Average Value of a Function?
The key formula is (1/(b−a))∫ f dx.
Where is Average Value of a Function used in real life?
Real-world examples: Finding the average temperature over a day from a continuous temperature curve.
What should I learn before Average Value of a Function?
Make sure you are comfortable with Surface Area of Revolution, the previous topic in Calculus.
What is a common mistake with Average Value of a Function?
Forgetting to divide by the length of the interval.
How can I practise Average Value of a Function?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Average Value of a Function?
Move on to Work and Fluid Force next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.