H: Integration: Understand and use integration as the limit of a sum.

The idea

Definite integration is defined as the limit of a sum of areas of rectangles. For a function f(x) on [a, b], divide the interval into n strips of width Δx = (b−a)/n. The sum of areas of rectangles, Σ f(x_i) Δx, approximates the area under the curve. As n → ∞ and Δx → 0, the sum tends to the definite integral ∫_a^b f(x) dx. This limit exists for continuous functions (and piecewise continuous). The notation ∫ comes from an elongated S for sum. For example, to approximate ∫_0^1 x^2 dx, use right endpoints: Σ_{i=1}^n (i/n)^2 (1/n) = (1/n^3) Σ i^2 = (1/n^3) n(n+1)(2n+1)/6 → 1/3 as n → ∞. This shows the integral equals 1/3. Understanding this limit underpins the Fundamental Theorem of Calculus and allows numerical methods like the trapezium rule to be seen as finite approximations.

What earns the marks

• Defines the definite integral as the limit of a sum of the form lim_{n→∞} Σ_{i=1}^n f(x_i) Δx, where Δx = (b−a)/n and x_i are sample points in each subinterval.
• Explains that the limit is taken as the number of subintervals increases without bound and the width of each subinterval tends to zero.
• Applies the definition to a simple function, such as f(x) = x or f(x) = x^2, by setting up the sum and evaluating the limit.
• Interprets the limit as the signed area under the curve y = f(x) between x = a and x = b, noting that areas below the x-axis contribute negatively.
• Connects the limit definition to the notation ∫_a^b f(x) dx and recognises that the integral is independent of the choice of sample points for continuous functions.

Common mistakes and how to put them right

• Treating Δx as a fixed small number rather than a quantity that tends to zero.

  Put it right: Emphasise that Δx = (b−a)/n and as n → ∞, Δx → 0.

• Using only left or right endpoints without considering the limit.

  Put it right: Show that for continuous functions, any choice of sample points gives the same limit, and practise with different sample points.

• Forgetting that the sum approximates signed area, so regions below the axis give negative contributions.

  Put it right: Illustrate with a function that crosses the x-axis and compute the sum to see negative terms.

• Assuming the limit always exists for any function.

  Put it right: Note that continuity (or piecewise continuity) is sufficient for the limit to exist; discuss a counterexample like the Dirichlet function if appropriate.

Exam tips

• When asked to 'use integration as the limit of a sum', set up the sum clearly, stating Δx and the sample points, then evaluate the limit using standard summation formulae.
• Show all steps in evaluating the limit, including algebraic manipulation of sums such as Σ i and Σ i^2, and state the limiting value explicitly.
• If a question asks for an interpretation, relate the limit to the area under the curve and mention that the integral gives the exact area as the number of strips tends to infinity.