A-Level · Mathematics · AQA · Mark scheme decoded

AQA A-Level Mathematics: Definite Integrals and Areas — mark scheme explained

Machine-verifiedchecked against the AQA A-Level Mathematics specificationlast verified 2 July 2026

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

What each mark is really for — in plain English — and the wording trap that loses it.

S1
Step 1: Find the antiderivative of f(x) = x 3 . The antiderivative is F(x) = (x 4 /4).
S2
Step 2: Evaluate F(2) and F(0). F(2) = (2 4 /4) = 16/4 = 4. F(0) = (0 4 /4) = 0.
S3
Step 3: Subtract the results to find the definite integral. Area = F(2) - F(0) = 4 - 0 = 4.

Model answer

Worked through, with each step tagged to the mark it earns.

S1
Step 1: Find the antiderivative of f(x) = x 3 . The antiderivative is F(x) = (x 4 /4).
S2
Step 2: Evaluate F(2) and F(0). F(2) = (2 4 /4) = 16/4 = 4. F(0) = (0 4 /4) = 0.
S3
Step 3: Subtract the results to find the definite integral. Area = F(2) - F(0) = 4 - 0 = 4.
Final answer: The area under the curve y = x 3 from x = 0 to x = 2 is 4 square units.

Common mistakes

Forgetting to take the absolute value of negative areas — 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 — 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 — Always evaluate the antiderivative at both the upper and lower limits and subtract the results.
Incorrectly splitting the integral when curves cross — Identify the points where the curves intersect and split the integral accordingly.
Using the wrong antiderivative — Double-check your antiderivative by differentiating it to ensure it matches the original function.
Forgetting to include the constant of integration in indefinite integrals — Always include +C when writing the antiderivative, even though it cancels out in definite integrals.

Where the marks go

Full worked solution (all marking points)3 marks