6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

Riemann Sums approximate area

A Riemann sum approximates the area under a curve by summing areas of rectangles: sum of f(x_i)·Δx across n subintervals. Left, right, and midpoint sums use the function value at the left endpoint, right endpoint, or midpoint of each subinterval to set rectangle height. As the number of rectangles increases (Δx → 0), the approximation improves.

Summation notation

Sigma notation, Σ (from i=1 to n) f(x_i)Δx, compactly represents adding up many rectangle areas. Each term uses a chosen x_i within subinterval i and the common width Δx = (b−a)/n. This notation is the bridge between finite approximations and the exact definite integral.

Definite integral as a limit

The definite integral ∫ₐᵇ f(x) dx is defined as the limit of Riemann sums as n → ∞ (Δx → 0): lim Σ f(x_i)Δx. This limit gives the exact signed area between f(x) and the x-axis from a to b, with area below the axis counted negative.

Key terms

Riemann sum

An approximation of area under a curve using the sum of rectangle areas: Σ f(x_i)Δx.

Left Riemann sum

Uses the left endpoint of each subinterval to determine rectangle height.

Midpoint Riemann sum

Uses the midpoint of each subinterval to determine rectangle height, often giving a more accurate estimate.

Δx (equal subintervals)

Width of each subinterval, calculated as (b − a)/n.

Definite integral ∫ₐᵇ f(x) dx

The limit of a Riemann sum as n → ∞; represents the exact signed area between f(x) and the x-axis from a to b.

Try an AP-style question

Which integral equals lim(n→∞) Σᵢ₌₁ⁿ (1 + 2i/n)² · (2/n)?

1. ∫₁³ x² dx
2. ∫₀³ x² dx
3. ∫₀² x² dx
4. ∫₁² x² dx