Part III — Integration

"Read Euler, read Euler, he is the master of us all." — Pierre-Simon Laplace

In Part II we learned to compute derivatives. Now we reverse the operation.

Integration is the second fundamental operation of calculus, and it answers a question that seems entirely unrelated to derivatives: what is the area under a curve? The shocking discovery, made independently by Newton in 1665 and Leibniz in 1675, is that integration and differentiation are inverse operations. The slope problem and the area problem — which mathematicians had treated as completely separate for two thousand years — turned out to be the same problem.

This is the Fundamental Theorem of Calculus. It is the single most important theorem in mathematics. Chapter 14 is the climax of this textbook's first half; everything before builds toward it, and everything after builds upon it.

What This Part Covers

  • Chapter 13 — The Definite Integral. The area problem. Riemann sums (left, right, midpoint). The definite integral as the limit of Riemann sums. Numerical integration with Python. First appearance of the area-under-the-normal-curve anchor.

  • Chapter 14 — The Fundamental Theorem of Calculus. The chapter of this book's first half. FTC Part 1 and Part 2, both proved and interpreted geometrically. Why this theorem is astonishing. Why it made modern science possible.

  • Chapter 15 — Integration Techniques I: Substitution and Integration by Parts. $u$-substitution (the reverse chain rule). Integration by parts (the reverse product rule). The two most important techniques you will use for the rest of your mathematical life.

  • Chapter 16 — Integration Techniques II. Trigonometric integrals, trigonometric substitution, partial fractions, numerical methods (trapezoidal rule, Simpson's rule).

  • Chapter 17 — Improper Integrals. When the limits stretch to infinity, or the integrand blows up. The Gamma function preview. The Gaussian integral $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$ (stated; proved in Chapter 32).

  • Chapter 18 — Applications of Integration. Area between curves. Volumes by disk, washer, and shell methods. Arc length, surface area, work, hydrostatic force, center of mass.

  • Chapter 19 — Differential Equations. Where calculus meets the real world. Separable equations. Direction fields. Euler's method. Applications: population growth (logistic), Newton's law of cooling, radioactive decay, mixing problems. First full development of the SIR model anchor.

What You Should Be Able to Do by the End of Part III

  • Compute definite integrals exactly using FTC
  • Apply $u$-substitution, integration by parts, partial fractions, and trig substitution fluently
  • Set up integrals for areas, volumes, arc lengths, and physical quantities (work, mass)
  • Recognize when an integral is improper and determine whether it converges
  • Solve simple first-order differential equations analytically
  • Use Python with sympy for symbolic integration and scipy for numerical integration and ODE solving
  • Build the SIR model and use it to simulate epidemic dynamics

Why This Part Matters

Differentiation tells you how things change. Integration tells you how those changes accumulate over time. Together they let you predict the future of any system you can model. That is why every quantitative discipline — physics, engineering, biology, economics, data science — depends on integration.

This part is also where the textbook makes good on its promise that calculus is for every field. The differential equations chapter (Chapter 19) models populations, diseases, cooling objects, and decaying radioisotopes. The applications chapter (Chapter 18) computes work done by varying forces, areas of irregular regions, and the center of mass of a non-uniform body. The Gaussian integral in Chapter 17 sets up everything you will ever do in probability and statistics.

If Part II was about building fluency, Part III is about building capability. You will leave it able to answer real questions about real systems. That is the entire point of calculus.

Chapters in This Part