MathematicsChapter 3: Integration
Average Value of A Function
Average value on [a,b] is (1/(b-a)) integral_a^b f(x)dx.
Learning objectives
- State the rule for average value of a function.
- Solve a average value of a function problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Average value on [a,b] is (1b-a) integralab f(x)dx.
Where / symbols
- Average value of a function
- Average value on [a,b] is (1/(b-a)) integral_a^b f(x)dx.
CONCEPT AND RULE
Average value on [a,b] is (1b-a) integralab f(x)dx.
METHOD
For average value of a function, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
Average of x2 on [0,3] is (13)[xfrac33]03=3.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve average value of a function
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- Average of x2 on [0,3] is (13)[xfrac33]03=3.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Average value of a function
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Average Value of A Function
Sticky Notes · Formula and Worked Revision
Core Idea
- Average value on [a,b] is (1b-a) integralab f(x)dx.
Rule / Formula
- Average value on [a,b] is (1b-a) integralab f(x)dx.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Average of x2 on [0,3] is (13)[xfrac33]03=3.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- Average value on [a,b] is (1b-a) integralab f(x)dx.
- Worked pattern: Average of x2 on [0,3] is (13)[xfrac33]03=3.
- Final check: domain, sign, notation, units, and reasonableness.
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