Chapter 14 — Further Reading

Where to go deeper on the Fundamental Theorem of Calculus. Each entry says what to read and why, with explicit section mapping to the two reference texts this book is benchmarked against (see _continuity.md §8). Full citations live at the bottom.

Standard Coverage — The Two Forms of FTC

  • Stewart, Calculus: Early Transcendentals (9th ed.), §5.3 "The Fundamental Theorem of Calculus." The canonical undergraduate treatment. Stewart presents FTC Part 1 and Part 2 in the same order we do, with the accumulation-function motivation and the Mean Value Theorem for Integrals as the engine of the proof. Work his Examples on differentiating $\int_a^{x} f$ and the chain-rule cases — they are the direct analog of our §14.8. Maps to our §14.2–§14.6, §14.8.
  • OpenStax Calculus Volume 1, §5.3 "The Fundamental Theorem of Calculus." Free and excellent. Slightly more leisurely than Stewart, with extra worked accumulation-function examples and a clean statement of the Mean Value Theorem for Integrals. Best first stop if the proof in our §14.3 Math Major Sidebar moved too fast. Maps to our §14.3–§14.4, §14.9.

Net Change, Displacement, and Distance

  • Stewart §5.4 "Indefinite Integrals and the Net Change Theorem." This is the home of our §14.7. Stewart works rectilinear-motion problems where you must distinguish net displacement $\int v\,dt$ from total distance $\int |v|\,dt$ — the single most-tested skill in the chapter. Do these before the chapter exam. Maps to our §14.7, §14.14 Error 3.
  • OpenStax Vol. 1, §5.4 "Integration Formulas and the Net Change Theorem." Parallel free coverage with additional applied net-change problems (cost, population, charge). Maps to our §14.7, §14.10.

Average Value and the Mean Value Theorem for Integrals

  • Stewart §6.5 "Average Value of a Function." Develops $\bar f = \frac{1}{b-a}\int_a^b f$ and the existence of a point where the average is attained, with applied examples (average temperature, average velocity). Maps to our §14.9.
  • OpenStax Vol. 1, §5.3 also states and proves the Mean Value Theorem for Integrals inline; read it for a second angle on why some $c$ achieves the average. Maps to our §14.9.

When Antiderivatives Are Not Elementary

  • Stewart §7.5 "Strategy for Integration" and §7.8 "Improper Integrals." Stewart discusses non-elementary integrands and points toward numerical and series methods; pair this with our §14.12. The improper-integral section is the rigorous home for the §14.4 pitfall integral $\int_{-1}^{1} x^{-2}\,dx$. Maps to our §14.12 and forward to Chapter 17.
  • OpenStax Vol. 2, §3.7 "Improper Integrals." Free treatment of the discontinuity-inside-the-interval trap and unbounded limits. Maps to our §14.4 pitfall, Chapter 17.

Applications (Probability, Pharmacokinetics, Work)

  • Stewart §8.5 "Probability." Density functions and CDFs as definite integrals — the calculus behind Case Study 2. Read it to see the mean and variance computed as the integrals we flagged as out of reach with §14 tools alone. Maps to our §14.10, Case Study 2.
  • OpenStax Vol. 1, §2.1 "Areas Between Curves" and Vol. 2, §1.1 "Approximating Areas" build the area-as-accumulation intuition that underlies our pharmacokinetic and AUC examples. Maps to our §14.10, Case Study 1.

The Big Picture — FTC as the Seed of Vector Calculus

  • Stewart §16.3 "The Fundamental Theorem for Line Integrals" (our Chapter 35), §16.4 "Green's Theorem" (Chapter 35), §16.8 "Stokes' Theorem" and §16.9 "The Divergence Theorem" (Chapter 37). Read the opening paragraph of each now — not to master them, but to recognize the §14.13 slogan returning: the integral of a derivative over a region equals values on the boundary. Maps to our §14.13.
  • Spivak, Calculus (4th ed.), ch. 13–14. For math majors who want FTC proved with full rigor, including the subtleties of which hypotheses are truly needed. Spivak's treatment of the Mean Value Theorem for Integrals and uniform continuity sharpens our §14.3 Math Major Sidebar. Maps to our §14.3 (formal level).

Historical Reading

  • Dunham, The Calculus Gallery, ch. on Newton and Leibniz. A readable account of the independent discovery of FTC and the priority dispute summarized in our §14.12 Historical Note. Explains why Leibniz's $\int$ and $dy/dx$ notation beat Newton's fluxions.
  • Boyer, The History of the Calculus and Its Conceptual Development. The deeper scholarly history, from Archimedes' areas through the seventeenth-century unification. For readers who want the two-thousand-year arc behind §14.1.

Full Citations

  • Stewart, James. Calculus: Early Transcendentals, 9th ed. Cengage, 2020.
  • Strang, Gilbert, and Edwin Herman. Calculus, Volumes 1–3. OpenStax, Rice University, 2016. Free at openstax.org.
  • Spivak, Michael. Calculus, 4th ed. Publish or Perish, 2008.
  • Ross, Sheldon. A First Course in Probability, 10th ed. Pearson, 2018.
  • Dunham, William. The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton University Press, 2005.
  • Boyer, Carl B. The History of the Calculus and Its Conceptual Development. Dover, 1959.

Chapter-mapping appendices. For the complete section-by-section correspondence between this book and the two reference texts, see appendices/appendix-h-stewart-chapter-mapping.md and appendices/appendix-i-openstax-chapter-mapping.md.