Source: Calc AB Lecture Notes, UCF Advanced Calculus
Difficulty: Intermediate | Prerequisites: Antiderivatives and indefinite integrals (Part 1 of these notes)
Tags: definite integral, Riemann sum, left sum, right sum, midpoint, trapezoidal rule, FTC, Fundamental Theorem of Calculus, area under a curve, properties of integrals
Where antiderivatives gave you families of functions, definite integrals give you numbers. This section connects the idea of "area under a curve" to the machinery of antidifferentiation. Riemann sums approximate that area using rectangles or trapezoids; the Fundamental Theorem of Calculus then shows you can skip the approximation entirely and compute exact areas using antiderivatives. This is the central bridge of the entire course, and virtually every applied integration problem from here on relies on it.
A definite integral ∫ from a to b of f(x) dx gives the net signed area between f(x) and the x-axis on [a, b]. You can approximate it with Riemann sums (rectangles) or the trapezoidal rule, but the Fundamental Theorem of Calculus lets you evaluate it exactly as F(b) − F(a), where F is any antiderivative of f.
Definite integral
The notation ∫ₐᵇ f(x) dx, where a is the lower limit and b is the upper limit. It evaluates to a specific number representing net signed area. Think of it as: "the total accumulation of f(x) from x = a to x = b."
Riemann sum
An approximation of a definite integral made by slicing the interval [a, b] into n subintervals and summing the areas of rectangles whose heights are sampled from f(x). In simple terms: "chop the area into rectangles and add them up."
Left, right, and midpoint rectangles
Three ways to choose the height of each rectangle. Left uses the function value at the left edge of each subinterval, right uses the right edge, and midpoint uses the centre.
Δx (delta x)
The width of each subinterval: Δx = (b − a) / n, where n is the number of subdivisions.
Trapezoidal rule
An approximation that uses trapezoids instead of rectangles, generally giving a better estimate. Formula for n subdivisions: [(b − a) / (2n)] · [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]. Think of it as: "connecting the dots with straight lines instead of flat tops."
Fundamental Theorem of Calculus (FTC)
∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f. This theorem says that integration and differentiation are inverse operations. In simple terms: "to find the exact area, just plug the limits into the antiderivative and subtract."
Net signed area
Area above the x-axis counts as positive; area below the x-axis counts as negative. The definite integral gives the net result (positive minus negative).
A definite integral represents the area under f(x) and above the x-axis on [a, b], minus any area below the x-axis.
To approximate this area, divide [a, b] into n equal subintervals, each of width Δx = (b − a) / n.
Three types of rectangular approximation:
Left Riemann sum: sample f at the left endpoint of each subinterval.
Right Riemann sum: sample f at the right endpoint.
Midpoint Riemann sum: sample f at the centre of each subinterval.
As n increases (more rectangles), all three methods converge to the true integral.
Uses trapezoids rather than rectangles for a typically more accurate estimate.
Area of a single trapezoid: ½(b₁ + b₂)h, where b₁ and b₂ are the two parallel sides (function values) and h is Δx.
Full formula for n subdivisions: [(b − a) / (2n)] · [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)].
Notice the pattern: first and last values appear once; all interior values are doubled.
∫ₐᵃ f(x) dx = 0. Integrating over a zero-width interval gives zero.
∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx. Swapping the limits negates the integral.
Area below the x-axis contributes negative value to the integral.
Constants factor out: ∫ₐᵇ k·f(x) dx = k · ∫ₐᵇ f(x) dx.
Sums and differences split: ∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx.
Note: you can only split sums and differences, not products or quotients.
Part 1 (evaluation): ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).
This eliminates the need for Riemann sums when you can find an antiderivative.
No + C required for definite integrals (it cancels in the subtraction).
Worked example:
∫₁² (x² − 3) dx
Antiderivative: F(x) = x³/3 − 3x
F(2) − F(1) = (8/3 − 6) − (1/3 − 3) = (−10/3) − (−8/3) = −2/3
Part 2 (derivative of an integral): d/dx ∫ₐˣ f(t) dt = f(x).
If the upper limit is a function of x (say g(x) instead of x), apply the chain rule: d/dx ∫ₐᵍ⁽ˣ⁾ f(t) dt = f(g(x)) · g'(x).
Worked example (chain rule version):
d/dx ∫₃ˣ² (eᵗ + sin t) dt = (eˣ² + sin(x²)) · 2x
Formula | Expression |
|---|---|
Subinterval width | Δx = (b − a) / n |
Trapezoidal rule (n subdivisions) | [(b − a) / (2n)] · [f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)] |
FTC Part 1 | ∫ₐᵇ f(x) dx = F(b) − F(a) |
FTC Part 2 | d/dx ∫ₐˣ f(t) dt = f(x) |
FTC Part 2 (chain rule) | d/dx ∫ₐᵍ⁽ˣ⁾ f(t) dt = f(g(x)) · g'(x) |
Riemann sums are how software computes integrals numerically when no clean antiderivative exists, which is most real engineering problems. The trapezoidal rule (and its refinements) power everything from calculating the area of an irregular plot of land to estimating total drug exposure in pharmacokinetics from blood-concentration data sampled at discrete time points.
Treating definite integrals as always positive. The integral ∫ₐᵇ f(x) dx can be negative if f(x) is below the x-axis for most or all of [a, b]. It gives net signed area, not geometric area.
Adding + C to a definite integral. The constant of integration cancels when you compute F(b) − F(a). Only indefinite integrals need + C.
Forgetting the chain rule in FTC Part 2. When the upper limit is g(x) rather than just x, you must multiply by g'(x). This is tested frequently.
Confusing the trapezoidal rule's coefficient pattern. The first and last function values get coefficient 1; all interior values get coefficient 2. Mixing this up changes the answer.
⚠️ FTC Part 1 evaluation problems appear on nearly every exam. You must be able to antidifferentiate and then evaluate at the limits cleanly.
⚠️ FTC Part 2 with the chain rule is a favourite multiple-choice question. Practise recognising when the upper limit is a function of x.
⚠️ Know how to set up and compute a trapezoidal approximation from a table of values. This is a standard free-response format.
⚠️ The property ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx is tested as a "trick" in multiple-choice. Read the limits carefully.
True or false: ∫₃³ f(x) dx = 0.
Fill in the blank: Swapping the limits of a definite integral ______ the sign.
True or false: A definite integral can never be negative.
Fill in the blank: d/dx ∫₀ˣ² sin(t) dt = ______.
True or false: The trapezoidal rule doubles the first and last function values.
Answers: 1. True. 2. Reverses (negates). 3. False. 4. sin(x²) · 2x. 5. False (it doubles the interior values; the first and last appear once).
Q: Evaluate ∫₀³ (2x + 1) dx using the Fundamental Theorem of Calculus.
A: Antiderivative: F(x) = x² + x. F(3) − F(0) = (9 + 3) − (0) = 12.
Q: A table gives f(0) = 2, f(1) = 5, f(2) = 4, f(3) = 6. Use the trapezoidal rule with n = 3 to approximate ∫₀³ f(x) dx.
A: Δx = 1. Trapezoidal sum = (1/2)[f(0) + 2f(1) + 2f(2) + f(3)] = (1/2)[2 + 10 + 8 + 6] = 13.
Q: Find d/dx ∫₂ˣ³ cos(t) dt.
A: cos(x³) · 3x² (apply FTC Part 2 with the chain rule).
Q: If ∫₁⁴ f(x) dx = 7, what is ∫₄¹ f(x) dx?
A: −7. Swapping the limits negates the integral.
Riemann sums formalise the intuition behind definite integrals and connect to the limit definition of the integral covered in theoretical calculus courses. The FTC links this section back to antiderivatives (Part 1 of these notes) and forward to applied accumulation and motion problems. The net-signed-area concept also connects to displacement versus total distance, covered in Part 3 of these notes.
definite integral, Riemann sum, left Riemann sum, right Riemann sum, midpoint approximation, trapezoidal rule, trapezoid approximation, Fundamental Theorem of Calculus, FTC, FTC Part 1, FTC Part 2, area under curve, net signed area, properties of integrals, Calc AB Unit 4, evaluation theorem, accumulation function