1. Why include C in an indefinite integral?
- Every curve passes through zero
- C is always 1
- Constants have derivative zero
Check the answer and reasoning
Constants have derivative zero
Differentiation loses the information about an additive constant.
MATH / AS level
Integration and area
Integration reverses differentiation and accumulates signed change. An indefinite integral represents a family of functions differing by a constant.
Learning objectives
- Explain integration and area using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Integration reverses differentiation and accumulates signed change. An indefinite integral represents a family of functions differing by a constant.
A definite integral is signed area. To find total geometric area, split the interval where the curve crosses the axis and handle negative pieces separately.
Equations and relationships
Reverse power rule.
Worked example
Evaluate ∫ from 0 to 2 of 3x² dx.
Follow the solution
An antiderivative is x³; evaluate 2³−0³=8.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.
Check your understanding
2. Evaluate ∫ from 0 to 2 of 3x² dx.
- 8
- 12
- 6
Check the answer and reasoning
8
An antiderivative is x³; evaluate 2³−0³=8.
3. Can a definite integral be negative?
- No
- Yes
- Only if both limits are negative
Check the answer and reasoning
Yes
Values below the axis contribute negative signed area regardless of the signs of the limits.