Chapter 13 — Further Reading

Everything below treats the pre-FTC definite integral: Riemann sums, the limit definition, signed area, properties, average value, and numerical integration. Each entry says what to read it for and which section of this chapter it matches. The two anchor textbooks — Stewart and OpenStax — are mapped section by section; a full cross-reference table lives in appendix-h-stewart-chapter-mapping.md and appendix-i-openstax-chapter-mapping.md.

Standard Coverage — The Two Anchor Texts

Stewart, Calculus: Early Transcendentals (9th ed., Cengage)

The book this text is benchmarked against for breadth and exercise depth (continuity tracker §8). For Chapter 13, read Chapter 5, Sections 5.1–5.2 and 5.5, plus 7.7:

  • §5.1 "Areas and Distances" — the area problem and the distance-from-velocity problem, developed exactly as our §13.1–13.2 and §13.9–13.10. Stewart's velocity-table example is the direct cousin of Case Study 2. Read for: the twin motivations (area and distance) that make Riemann sums feel inevitable.
  • §5.2 "The Definite Integral" — the limit-of-Riemann-sums definition, sigma notation, the properties, and the Midpoint Rule. Matches our §13.3–13.8. Read for: the canonical statement of the definition and a thorough properties list with proofs.
  • §5.5 "The Substitution Rule" — skip for now; it depends on the FTC (Chapter 14 here). Listed only so you do not wander into it expecting pre-FTC material.
  • §7.7 "Approximate Integration" — the Midpoint, Trapezoidal, and Simpson's rules with error bounds. Matches our §13.13 (Simpson's rule itself is our Chapter 16). Read for: the rigorous error analysis behind the convergence you watched in both case studies.

Note: Stewart introduces the Fundamental Theorem in §5.3. Stop before it if you want to stay in the spirit of this chapter, which deliberately evaluates integrals without the shortcut.

OpenStax, Calculus Volume 1 (Strang & Herman, free)

The open-access text this book aims to exceed in application breadth (continuity tracker §8). For Chapter 13, read Chapter 5, Sections 5.1–5.2:

  • §5.1 "Approximating Areas" — left, right, and midpoint Riemann sums, sigma notation, and the sum formulas $\sum i, \sum i^2, \sum i^3$. Matches our §13.2–13.3 and §13.5. Read for: abundant worked sums and a gentle build-up to the limit. Free, so this is the place to drill mechanics.
  • §5.2 "The Definite Integral" — the limit definition, signed area, the properties, and average value with the Mean Value Theorem for Integrals. Matches our §13.4, §13.6, §13.8, and §13.11. Read for: a second pass over the properties and a clean statement of the MVT for Integrals.

OpenStax presents the FTC in §5.3; like Stewart's §5.3, save it for our Chapter 14.

Deeper and More Rigorous

Spivak, Calculus (4th ed., Publish or Perish)

The proof-heavy reference for conceptual clarity (continuity tracker §8). Chapter 13 "Integrals" builds the Riemann integral from upper and lower (Darboux) sums and proves that continuous functions are integrable — the rigorous version of our §13.12 and the Math Major Sidebar in §13.4. Read for: the genuine existence proof and the upper/lower-sum machinery, if you want the analysis behind the squeeze argument.

Apostol, Calculus, Volume I (2nd ed., Wiley)

Develops the integral before the derivative, the reverse of most texts, emphasizing the integral as an independent object. Read for: a fresh perspective on why the integral deserves to be built on its own terms — exactly the stance this chapter takes by refusing to lean on the FTC.

Numerical Integration in Practice

Burden & Faires, Numerical Analysis — Chapter 4 "Numerical Differentiation and Integration"

The engineering-standard treatment of the trapezoidal rule, midpoint rule, and their error terms, extending our §13.13. Read for: derivations of the $O(h^2)$ error bounds and a preview of composite and adaptive rules.

SciPy documentation — scipy.integrate (quad, trapezoid, simpson)

The production tools that generalize the hand-coded Riemann sums in both case studies. Read for: how real software evaluates an integral like the bell-curve area of §13.14 to machine precision — the grown-up version of the loops you saw.

The Normal-Curve Anchor (§13.14)

OpenStax, Introductory Statistics — "The Normal Distribution"

The applied destination of Case Study 1: how the area $\int_a^b \phi$ becomes a probability, a $z$-score, and the "68–95–99.7" rule, with no calculus assumed. Read for: where the number $0.6827$ goes once you have computed it.

Stigler, The History of Statistics (Harvard, 1986)

The story of how the normal curve became central to science — Gauss, Laplace, and the method of least squares. Read for: historical depth on the function this chapter adopts as its through-line anchor (continued in Chapters 14 and 23).

A Note on Reading Order

If you read only one outside source, make it OpenStax §5.1–5.2 — free, complete for this chapter, and rich in practice. If you want rigor, add Spivak Chapter 13. If your interest is applied, pair Stewart §7.7 with the SciPy docs and run the case-study code yourself. In every case, resist peeking at the Fundamental Theorem (anyone's §5.3) until Chapter 14 — the payoff is far greater once you have felt how hard the integral is to compute by hand.