AQA A-Level Mathematics: Integration as the Limit of a Sum — mark scheme explained

The question

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.

Model answer

A full-mark response to the question above, worked through step by step.

⏱Timing: Allocate about 5-7 minutes for a question involving Riemann sums and 3-5 minutes for a question involving direct integration.

1. Divide the interval [0, 2] into 4 subintervals of equal width: Δx = (2 - 0) / 4 = 0.5.1 mark
2. The subintervals are [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2].0 marks
3. Choose the right endpoints as sample points: x 1 * = 0.5, x 2 * = 1, x 3 * = 1.5, and x 4 * = 2.1 mark
4. Form rectangles with heights f(x i * ) = (x i * ) 2 :0 marks
5. f(0.5) = (0.5) 2 = 0.25, f(1) = 1 2 = 1, f(1.5) = (1.5) 2 = 2.25, and f(2) = 2 2 = 4.1 mark
6. The areas of the rectangles are: 0.25 × 0.5 = 0.125, 1 × 0.5 = 0.5, 2.25 × 0.5 = 1.125, and 4 × 0.5 = 2.2 marks
7. Sum the areas of the rectangles: S = 0.125 + 0.5 + 1.125 + 2 = 3.75.1 mark

Final answer: The approximate area under the curve y = x 2 on [0, 2] using a Riemann sum with n = 4 subintervals and right endpoints is 3.75.

✓Work through every step correctly and you earn all 6 marks.

Common mistakes

• Forgetting to divide the interval into subintervals of equal width.

  Why it happens: Students sometimes forget to ensure that each subinterval has the same width, which is crucial for a consistent Riemann sum approximation.

  Fix: Always calculate Δx = (b - a) / n and use this value consistently for all subintervals.

• Choosing incorrect sample points in the subintervals.

  Why it happens: Students may choose sample points that are not at the left endpoint, right endpoint, or midpoint of each subinterval, leading to inaccurate approximations.

  Fix: Clearly specify which type of sample point (left, right, or midpoint) you are using and stick to it consistently.

• Failing to sum the areas of all rectangles correctly.

  Why it happens: Students might make arithmetic errors when summing the areas of the rectangles, leading to incorrect Riemann sums.

  Fix: Double-check your calculations and ensure that you are summing the areas of all rectangles accurately.

• Not taking the limit as n approaches infinity when finding the exact area.

  Why it happens: Students sometimes forget to take the limit of the Riemann sum as n approaches infinity, which is necessary to find the exact area using integration.

  Fix: Always remember that the exact area under the curve is given by the definite integral ∫ a b f(x) dx, which is the limit of the Riemann sum as n → ∞.

• Using incorrect formulas for sums of series in Riemann sums.

  Why it happens: Students might use incorrect or incomplete formulas for sums of series, leading to errors in the Riemann sum approximation.

  Fix: Review and memorize the correct formulas for sums of series, such as Σ i=1 n i = n(n + 1)/2 and Σ i=1 n i 2 = n(n + 1)(2n + 1)/6.

• Forgetting to evaluate the definite integral at both bounds.

  Why it happens: Students might forget to substitute both the upper and lower bounds into the antiderivative when evaluating a definite integral, leading to incorrect results.

  Fix: Always evaluate the antiderivative at both the upper and lower bounds of the interval and subtract the results to find the exact area under the curve.

• Confusing the width of subintervals with the number of subintervals.

  Why it happens: Students might confuse Δx (the width of each subinterval) with n (the number of subintervals), leading to errors in their calculations.

  Fix: Clearly distinguish between Δx and n. Δx is the width of each subinterval, while n is the total number of subintervals.

• Using incorrect limits of integration.

  Why it happens: Students might use the wrong bounds when setting up the definite integral, leading to incorrect results.

  Fix: Always double-check the interval [a, b] and ensure that you are using the correct bounds in your definite integral.

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 types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.

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