MathematicsChapter 3: Integration
Area Under A Curve
Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
Learning objectives
- State the rule for area under a curve.
- Solve a area under a curve problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
Mathematical form
y=f(x) and x-axis is integral f(x)dx where
y=x from 0 to 3 is 92.
Where / symbols
- Area under a curve
- Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
CONCEPT AND RULE
Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
METHOD
For area under a curve, 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
Area under y=x from 0 to 3 is 92.
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 area under a curve
- 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:
- Area under y=x from 0 to 3 is 92.
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
- Area under a curve
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Area Under A Curve
Sticky Notes · Formula and Worked Revision
Core Idea
- Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
Rule / Formula
- Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Area under y=x from 0 to 3 is 92.
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
- Area between y=f(x) and x-axis is integral f(x)dx where f>=0; split where sign changes.
- Worked pattern: Area under y=x from 0 to 3 is 92.
- Final check: domain, sign, notation, units, and reasonableness.
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