Source: Chapter 5, Section 5.1 | Calculus (Texas A&M University)
Tags: Riemann sum, definite integral, area under a curve, sigma notation, partition, subinterval, right endpoint, left endpoint, sample point, signed area, integral properties, summation formulas
The definite integral is defined as the limit of Riemann sums, which approximate the area under a curve by slicing it into thin rectangles. For continuous functions, the limit exists regardless of how you choose your sample points. In practice, you rarely compute integrals via Riemann sums directly, but the concept is the foundation for setting up integrals.
Riemann sum
The expression ∑ f(xᵢ*) Δx, formed by partitioning [a, b] into n subintervals of width Δx = (b − a)/n, choosing a sample point xᵢ* in each subinterval, and summing the products of function value times width. Approximates the definite integral.
Partition
A subdivision of [a, b] into n subintervals using points a = x₀ < x₁ < ... < xₙ = b. In this course, the partition is usually taken to be equally spaced (uniform partition).
Sample point (xᵢ)*
A point chosen inside the i-th subinterval [xᵢ₋₁, xᵢ]. Common choices are the left endpoint, right endpoint, or midpoint. The definition of the definite integral guarantees the limit is the same for any valid choice.
Definite integral
∫ₐᵇ f(x) dx. The limit of Riemann sums as n → ∞. For a continuous function on [a, b], this limit always exists and equals a single fixed number, regardless of the sample points chosen.
Signed area
The definite integral gives signed area: regions above the x-axis contribute positively, regions below contribute negatively. If f(x) ≥ 0 on [a, b], the definite integral equals the geometric area.
Δx (delta x)
The width of each subinterval in a uniform partition: Δx = (b − a)/n.
The motivating idea is to find the area under y = f(x) from x = a to x = b.
Divide [a, b] into n equal subintervals, each of width Δx = (b − a)/n.
In each subinterval, pick a sample point xᵢ* and form a rectangle of height f(xᵢ*) and width Δx.
The Riemann sum is the total area of these rectangles: ∑ᵢ₌₁ⁿ f(xᵢ*) Δx.
The "fence" analogy from the notes: think of the area under f(x) = 2 + sin x on [0, 2π] as a fence made of boards. Each board is approximately a rectangle. The total fence area is the sum of the board areas, and as you use more, thinner boards, the approximation improves.
Let f be continuous on [a, b]. Set Δx = (b − a)/n. For any choice of sample points xᵢ* (one per subinterval), the Riemann sum ∑ᵢ₌₁ⁿ f(xᵢ*) Δx approaches a single fixed number as n → ∞. That number is the definite integral:
∫ₐᵇ f(x) dx = lim (n→∞) ∑ᵢ₌₁ⁿ f(xᵢ*) Δx
The key point: continuity of f on [a, b] guarantees the limit exists and is the same no matter how you pick the sample points.
Constant multiple: ∫ₐᵇ c·f(x) dx = c · ∫ₐᵇ f(x) dx
Sum rule: ∫ₐᵇ [f(x) + g(x)] dx = ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx
Interval splitting: If a ≤ b ≤ c, then ∫ₐᶜ f(x) dx = ∫ₐᵇ f(x) dx + ∫ᵇᶜ f(x) dx
These mirror the linearity rules you already know from derivatives. They let you break complicated integrals into simpler pieces.
∫₀ᵇ 1 dx = b
The function f(x) = 1 is a horizontal line at height 1. The area is just a rectangle of width b and height 1.
∫₀ᵇ x dx = b²/2
Geometrically, the area under the line y = x from 0 to b is a triangle with base b and height b.
To verify with Riemann sums (right endpoints): Δx = b/n, xᵢ* = ib/n.
∑ᵢ₌₁ⁿ (ib/n)(b/n) = (b²/n²) ∑ᵢ₌₁ⁿ i = (b²/n²) · n(n+1)/2 = (b²/2) · (n+1)/n → b²/2
∫₀ᵇ x² dx = b³/3
No simple geometric shortcut here. With right endpoints: Δx = b/n, xᵢ* = ib/n.
∑ᵢ₌₁ⁿ (ib/n)²(b/n) = (b³/n³) ∑ᵢ₌₁ⁿ i² = (b³/n³) · n(n+1)(2n+1)/6 → b³/3
The takeaway: computing integrals this way is tedious. The Fundamental Theorem of Calculus (Section 5.2) gives a far better method.
Summation formulas you need for Riemann sum calculations:
∑ᵢ₌₁ⁿ i = n(n+1)/2
∑ᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6
Riemann sum setup (general):
Δx = (b − a)/n, xᵢ* ∈ [xᵢ₋₁, xᵢ], Riemann sum = ∑ᵢ₌₁ⁿ f(xᵢ*) Δx
⚠️ The definite integral is a number (signed area), not a function. This distinction matters when you get to FTC and indefinite integrals.
⚠️ Signed area: if f dips below the x-axis, those parts subtract from the total. Students often forget this when asked "find the area" versus "evaluate the integral."
⚠️ Riemann sum problems on exams usually ask you to set up and simplify the sum, then take a limit. You need the summation formulas for ∑ i and ∑ i² memorised.
⚠️ The integral ∫₃³ x sin x dx = 0 (same upper and lower limit). This is a common trick question.
Q: What is the width Δx when computing a Riemann sum for ∫₁⁴ f(x) dx with n subintervals?
A: Δx = (4 − 1)/n = 3/n.
Q: Using right endpoints and n subintervals, write the Riemann sum for ∫₀³ x² dx.
A: Δx = 3/n, xᵢ* = 3i/n. The Riemann sum is ∑ᵢ₌₁ⁿ (3i/n)² · (3/n) = (27/n³) ∑ᵢ₌₁ⁿ i².
Q: Evaluate the limit of the Riemann sum above as n → ∞.
A: (27/n³) · n(n+1)(2n+1)/6 → 27 · (1/3) = 9. So ∫₀³ x² dx = 9.
Q: True or false: the definite integral ∫ₐᵇ f(x) dx is always positive.
A: False. The definite integral gives signed area. If f(x) < 0 on part or all of [a, b], the integral can be zero or negative.
Q: Why does ∫₃³ x sin x dx = 0?
A: When the upper and lower limits are equal, the interval has zero width, so the integral is 0.
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