Average Value of a Function

Average Value of a Function

What is it?

Average value on [a,b] is (1b-a) integral_a^b f(x)dx.

In simple words

• Average value on [a,b] is (1b-a) integral_a^b f(x)dx.
• Worked pattern: Average of x² on [0,3] is (13)[x^frac33]₀³=3.
• Final check: domain, sign, notation, units, and reasonableness.

Important terms

Average value of a function: Average value on [a,b] is (1b-a) integral_a^b f(x)dx.

What you should learn

• 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.

Full school-level explanation

Average value on [a,b] is (1b-a) integral_a^b f(x)dx.

CONCEPT AND RULE

Average value on [a,b] is (1b-a) integral_a^b 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)[x^frac33]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

How it works

• 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.

Remember

• Average value of a function
• rule
• worked example
• worked example

How the ideas connect

Worked example: Average of x² on [0,3] is (13)[x^frac33]₀³=3.

Deeper understanding

Go beyond memorising Average Value of a Function. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked example: Average of x² on [0,3] is (13)[x^frac33]₀³=3.

Common mistakes

Exam focus

Core Idea

Rule / Formula

Method

Worked Example

Exam focus

Quick revision

• Average value of a function
• rule
• worked example
• Average value of a function: Average value on [a,b] is (1b-a) integral_a^b f(x)dx.
• CONCEPT AND RULE: Average value on [a,b] is (1b-a) integral_a^b f(x)dx.
• METHOD: For average value of a function, identify the given information and required result. Select the matc

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

For long answers on Average Value of a Function: define clearly → explain the main parts → give one example → state one exam caution.