MathematicsChapter 3: Integration
Area Between Curves
Integrate upper minus lower after finding intersections and checking which graph is above.
Learning objectives
- State the rule for area between curves.
- Solve a area between curves problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Integrate upper minus lower after finding intersections and checking which graph is above.
Mathematical form
y=x and y
area=integral01(x-x2)dx
Where / symbols
- Area between curves
- Integrate upper minus lower after finding intersections and checking which graph is above.
CONCEPT AND RULE
Integrate upper minus lower after finding intersections and checking which graph is above.
METHOD
For area between curves, 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
Between y=x and y=x2 on [0,1], area=integral01(x-x2)dx=16.
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 between curves
- 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:
- Between y=x and y=x2 on [0,1], area=integral01(x-x2)dx=16.
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 between curves
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Area Between Curves
Sticky Notes · Formula and Worked Revision
Core Idea
- Integrate upper minus lower after finding intersections and checking which graph is above.
Rule / Formula
- Integrate upper minus lower after finding intersections and checking which graph is above.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Between y=x and y=x2 on [0,1], area=integral01(x-x2)dx=16.
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
- Integrate upper minus lower after finding intersections and checking which graph is above.
- Worked pattern: Between y=x and y=x^2 on [0,1], area=integral_0^1(x-x^2)dx=16.
- Final check: domain, sign, notation, units, and reasonableness.
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