Integration: 8.4 Understand and use integration as the limit of a sum.

The idea

Integration as the limit of a sum means the definite integral is defined by splitting the interval [a, b] into n strips of width Δx = (b − a)/n, forming the sum Σ f(xᵢ)Δx, and taking the limit as n → ∞ and Δx → 0. For example, for f(x) = x² on [0, 1], the right-endpoint sum is Σ (i/n)²(1/n) = (1/n³)Σ i² = (1/n³)·n(n + 1)(2n + 1)/6, which tends to 1/3, so ∫₀¹ x² dx = 1/3. This limit definition explains why integration gives signed area and why the definite integral is a number, while the indefinite integral is a family of functions. It also links to the fundamental theorem: differentiation and integration are inverse processes.

What earns the marks

• Defines a partition of [a, b] into n subintervals of equal width Δx = (b − a)/n and identifies a sample point xᵢ in each subinterval.
• Forms the Riemann sum Σ f(xᵢ)Δx and explains that it approximates the signed area under y = f(x) between x = a and x = b.
• Takes the limit as n → ∞ (equivalently Δx → 0) and states that the definite integral is the value of this limit when it exists.
• Uses the limit definition to evaluate a simple integral, such as ∫₀¹ x² dx = 1/3, by summing squares and letting n → ∞.
• Interprets the definite integral as a signed area, distinguishing regions below the x-axis from those above.
• Connects the limit-of-a-sum definition to the fundamental theorem of calculus, showing that integration reverses differentiation.

Common mistakes and how to put them right

• Treating Δx as a fixed finite width rather than a quantity that tends to zero.

  Put it right by stating Δx = (b − a)/n and taking the limit as n → ∞.

• Omitting the factor Δx when forming the sum.

  Put it right by writing each term as f(xᵢ)Δx before summing.

• Assuming every function has a limit of sums.

  Put it right by noting the definition applies when the limit exists, for example for continuous functions on a closed interval.

• Confusing the definite integral with the indefinite integral.

  Put it right by stating that the limit of a sum produces a number, whereas the indefinite integral produces a family of functions.

Exam tips

• When asked to explain integration as a limit, define Δx, write the sum, and state the limit in symbols before giving a numerical example.
• Use a simple function such as f(x) = x² on [0, 1] to demonstrate the method; show the algebra for Σ i² and the limiting value.
• If a question asks for the limit of a sum, check whether the interval is [a, b] and adjust Δx and the sample points accordingly.
• Link the limit definition to the fundamental theorem when explaining why ∫ₐᵇ f(x) dx can be evaluated using an antiderivative.