Integrals

An integral builds a whole back up out of infinitely many pieces. This course textbook starts with rectangles under a curve and does not stop until that one idea is computing volumes, arc lengths, work, and probability.

Unit 1, Foundations of Integrals, is the core story: approximating area, antiderivatives, the definite integral, the Fundamental Theorem of Calculus, and the Net Change Theorem.

Unit 2, Integration Rules, is the toolkit: integral properties and the average value, substitution, trigonometric integrals and substitution, and the exponential and logarithmic integrals.

Unit 3, Applications of Integrals, is seven ways the idea pays off: arc length, areas between curves, surface area of a revolution, volume methods, mass and density, force and work, and probability.

It is built to be worked, not just read. Each section opens with a concrete situation and carries the idea all the way through, then hands you practice backed by 546 hints you can open one at a time when you are stuck, and 342 fully worked solutions for when you want to check the whole chain. Behind the course sit two appendices: a ten-part prerequisite toolkit covering the trig, logs, geometry, physics, and probability the applications lean on, and a study-tools set with the notation, glossary, methods, formulas, and rules gathered in one place.

Accessibility is built in, not bolted on. All 11,842 mathematical expressions are real MathML carrying a spoken-English label, so VoiceOver reads a formula as a sentence instead of spelling it out symbol by symbol. All 90 instructional diagrams carry a full written description as well as short alternative text, reachable from the figure itself, so a reader who cannot see a diagram can still rebuild it. Text is selectable and resizable, navigation is structural, and the book does not override your reading theme, so White, Sepia, Gray and Night all work as you set them.

The twelve interactive widgets appear here as still images with descriptive captions. The live versions run in the free web edition at integrals.megan-warren.com.

The second of a two-volume pair with the Derivatives book. Written for grades 11 to 12. 2026-2027 edition.