A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Fundamental Theorem of Calculus — mark scheme explained
The short answer
The Fundamental Theorem of Calculus (FTC) is a cornerstone of calculus that links the concept of differentiation and integration. It consists of two parts, each providing a different perspective on how these operations are related.
The question
Evaluate ∫ 0 3 (2 x + 1) d x using Part 2 of the FTC.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions involving the FTC, marks are typically awarded for correctly identifying the antiderivative, applying the limits of integration, and providing the final answer. Partial credit may be given for correct steps even if the final answer is incorrect.
What the command words demand
- Evaluate
- Calculate the value of a definite integral using Part 2 of the FTC.
- Find
- Determine an antiderivative of a function and use it to solve a problem.
- Apply
- Use the Fundamental Theorem of Calculus to solve a given problem.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 3-5 minutes per question to ensure you have enough time to carefully apply the FTC and check your work.
- Find an antiderivative F ( x ) of f ( x ) = 2 x + 1.1 mark
- F ( x ) = ∫ (2 x + 1) d x = x 2 + x + C.1 mark
- Apply Part 2 of the FTC: ∫ 0 3 (2 x + 1) d x = F (3) - F (0).1 mark
- F (3) = 3 2 + 3 = 9 + 3 = 12.1 mark
- F (0) = 0 2 + 0 = 0.0 marks
- Therefore, ∫ 0 3 (2 x + 1) d x = 12 - 0 = 12.1 mark
Final answer: 12
Work through every step correctly and you earn all 5 marks.
Another worked example
If F ( x ) = ∫ 1 x (4 t - 3) d t , find F '( x ).
- By Part 1 of the FTC, if F ( x ) = ∫ 1 x (4 t - 3) d t , then F '( x ) = 4 x - 3.1 mark
- Therefore, F '( x ) = 4 x - 3.1 mark
Final answer: 4 x - 3
Work through every step correctly and you earn all 2 marks.
Common mistakes
Forgetting to apply the limits of integration in Part 2 of the FTC.
Why it happens: Students often find an antiderivative but forget to evaluate it at the upper and lower limits of integration.
Fix: Always remember to substitute the upper limit into the antiderivative, then subtract the value obtained by substituting the lower limit.
Confusing Part 1 and Part 2 of the FTC.
Why it happens: Part 1 deals with differentiation of an integral function, while Part 2 deals with evaluating definite integrals using antiderivatives. Students sometimes mix these up.
Fix: Review the statements of both parts of the FTC to understand their distinct applications.
Forgetting the constant of integration when finding an antiderivative.
Why it happens: In definite integrals, the constant of integration cancels out. However, in indefinite integrals, it is crucial to include it.
Fix: Always include the constant of integration C when finding an antiderivative unless you are evaluating a definite integral.
Incorrectly applying the FTC to non-continuous functions.
Why it happens: The FTC requires the function to be continuous on the interval of integration. Students sometimes apply it without checking this condition.
Fix: Always verify that the function is continuous on the given interval before applying the FTC.
Forgetting to check if a function has an antiderivative.
Why it happens: Not all functions have elementary antiderivatives. Students sometimes assume that every function can be integrated using basic techniques.
Fix: Be aware of the types of functions that do and do not have elementary antiderivatives, and use appropriate methods or numerical integration when necessary.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Definite Integral Evaluation | Find an antiderivative, apply the limits using the FTC, and compute the final value. | 5 |
| Apply Fundamental Theorem | Differentiate an integral with a variable upper limit using the Fundamental Theorem of Calculus. | 2 |
| Definite Integral Evaluation | Evaluate a definite integral by finding the antiderivative and applying the given limits. | 5 |
| Fundamental Theorem Calculus | Differentiate an integral with a variable upper limit to find F'(x). | 2 |
| Total across these question types | 14 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.