A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Integration as the Limit of a Sum — mark scheme explained
The short answer
Integration can be understood as the limit of a sum, which is a fundamental concept in calculus. This idea connects the process of integration to the more intuitive notion of adding up small quantities to find a total. Let's explore this concept step by step.
The question
Approximate the area under the curve y = x 2 on the interval [0, 2] using a Riemann sum with n = 4 subintervals and right endpoints. [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
Divide the interval [0, 2] into 4 subintervals of equal width: Δx = (2 - 0) / 4 = 0.5.
- S2
The subintervals are [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2].
- S3
Choose the right endpoints as sample points: x 1 * = 0.5, x 2 * = 1, x 3 * = 1.5, and x 4 * = 2.
- S4
Form rectangles with heights f(x i * ) = (x i * ) 2 :
- S5
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.
- S6
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.
- S7
Sum the areas of the rectangles: S = 0.125 + 0.5 + 1.125 + 2 = 3.75.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Divide the interval [0, 2] into 4 subintervals of equal width: Δx = (2 - 0) / 4 = 0.5.
- S2
The subintervals are [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2].
- S3
Choose the right endpoints as sample points: x 1 * = 0.5, x 2 * = 1, x 3 * = 1.5, and x 4 * = 2.
- S4
Form rectangles with heights f(x i * ) = (x i * ) 2 :
- S5
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.
- S6
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.
- S7
Sum the areas of the rectangles: S = 0.125 + 0.5 + 1.125 + 2 = 3.75.
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.
Common mistakes
- Forgetting to divide the interval into subintervals of equal width. — Always calculate Δx = (b - a) / n and use this value consistently for all subintervals.
- Choosing incorrect sample points in the subintervals. — 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. — 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. — 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. — 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. — 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. — 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. — Always double-check the interval [a, b] and ensure that you are using the correct bounds in your definite integral.
Where the marks go
- Full worked solution (all marking points)6 marks