A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Definite Integrals and Areas — mark scheme explained
The short answer
Integration is a fundamental concept in calculus that allows us to find the area under a curve or between two curves. In this section, we will focus on evaluating definite integrals and using them to calculate areas. Evaluating Definite Integrals A definite integral is an integral with specific limits of integration.
The question
Find the area under the curve y = x 3 from x = 0 to x = 2.
[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 definite integrals and areas, marks are typically awarded for correctly setting up the integral (1-2 marks), finding the antiderivative (1-2 marks), evaluating at the limits (1-2 marks), and providing the final answer (1 mark).
What the command words demand
- Evaluate
- Calculate the value of a definite integral.
- Find
- Determine the area under a curve or between two curves.
- Calculate
- Compute the exact value of an integral or area.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes per question involving definite integrals and areas, depending on the complexity of the problem.
- Find the antiderivative of f(x) = x 3 . The antiderivative is F(x) = (x 4 /4).1 mark
- Evaluate F(2) and F(0). F(2) = (2 4 /4) = 16/4 = 4. F(0) = (0 4 /4) = 0.1 mark
- Subtract the results to find the definite integral. Area = F(2) - F(0) = 4 - 0 = 4.1 mark
Final answer: The area under the curve y = x 3 from x = 0 to x = 2 is 4 square units.
Work through every step correctly and you earn all 3 marks.
Another worked example
Find the area between the curves y = x 2 and y = 2x from x = 0 to x = 2.
- Set up the integral for the area between the curves. Area = ∫ 0 2 (2x - x 2 ) dx.1 mark
- Find the antiderivative of (2x - x 2 ). The antiderivative is F(x) = (x 2 ) - (x 3 /3).1 mark
- Evaluate F(2) and F(0). F(2) = (2 2 ) - (2 3 /3) = 4 - 8/3 = 4/3. F(0) = (0 2 ) - (0 3 /3) = 0.1 mark
- Subtract the results to find the definite integral. Area = F(2) - F(0) = 4/3 - 0 = 4/3.1 mark
Final answer: The area between the curves y = x 2 and y = 2x from x = 0 to x = 2 is 4/3 square units.
Work through every step correctly and you earn all 4 marks.
Common mistakes
Forgetting to take the absolute value of negative areas
Why it happens: When calculating the total area under a curve, if parts of the curve are below the x-axis, the integral will give a negative result. For the total unsigned area, you need to take the absolute value of these segments.
Fix: Always check for negative areas and take their absolute values before summing them up.
Incorrectly setting up the integral for the area between two curves
Why it happens: Students often set up the integral as ∫(f(x) + g(x)) dx instead of ∫(f(x) - g(x)) dx, leading to incorrect results.
Fix: Always subtract the function that is lower from the one that is higher when setting up the integral for the area between two curves.
Forgetting to evaluate the antiderivative at both limits
Why it happens: Students sometimes only evaluate the antiderivative at one limit and forget to subtract the value at the other limit, leading to incorrect results.
Fix: Always evaluate the antiderivative at both the upper and lower limits and subtract the results.
Incorrectly splitting the integral when curves cross
Why it happens: When curves cross each other within the interval, students may not split the integral into multiple parts where one function is greater than the other, leading to incorrect areas.
Fix: Identify the points where the curves intersect and split the integral accordingly.
Using the wrong antiderivative
Why it happens: Students may use an incorrect antiderivative, leading to incorrect results when evaluating the definite integral.
Fix: Double-check your antiderivative by differentiating it to ensure it matches the original function.
Forgetting to include the constant of integration in indefinite integrals
Why it happens: While not necessary for definite integrals, students may forget that the antiderivative is a family of functions and should include +C.
Fix: Always include +C when writing the antiderivative, even though it cancels out in definite integrals.
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 Area | Integrate a power function and evaluate between limits to find the area under the curve. | 3 |
| Area Between Curves | Integrate the difference of two curves between given limits to find the enclosed area. | 4 |
| Area Under Curve | Find the total area between a curve and the x-axis, accounting for regions below the axis. | 5 |
| Definite Integral Area | Integrate the exponential function between two limits to find the area under the curve. | 3 |
| Total across these question types | 15 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.