I. Introduction
After Calculus I introduces integrals as accumulation, Calculus II (Integral Calculus) focuses on how to compute harder integrals with new techniques, and how to use integrals to solve geometry/physics-style problems. Courses usually finish with sequences and series.
II. Outline
- Integration Techniques
- Integration review + substitution
- Integration by parts
- Trigonometric integrals
- Trig substitution
- Partial fractions
- Improper integrals
- Numerical integration (Trapezoid / Simpson)
- Applications of Integration
- Area between curves
- Volumes of solids
- Volumes by slicing / known cross-sections
- Arc length and surface area of revolution (often later in the course)
- Work, fluid force, and other density/rate problems
- Average value of a function
- Parametric Equations & Polar Coordinates
- Parametric curves: derivatives, tangent lines, area, arc length
- Polar coordinates: graphs in polar form, area in polar, arc length
- Converting between rectangular ↔ polar
- Sequences & Series
- Sequences: convergence/divergence, monotone bounded behavior
- Infinite series basics: geometric & telescoping series
- Convergence tests
- Comparison / Limit Comparison
- Integral test
- Alternating series test
- Ratio test / Root test
- Power series + radius/interval of convergence
- Taylor / Maclaurin series: building series, approximations, error/remainder ideas
- Using series to evaluate limits, compute integrals, and approximate functions
III. Free Books
- OpenStax — Calculus Volume 2
- APEX Calculus — Downloads
- Scroll to APEX Calculus 2
- Active Calculus (Single Variable)
- Paul’s Online Notes — Calculus II
IV. Video Series
- Professor Leonard — Calculus 2 (Full Length Videos)
- freeCodeCamp / Dr. Linda Green — Calculus 2 Playlist
- Khan Academy — Calculus 2
- MIT OpenCourseWare — 18.01SC Single Variable Calculus (strong for integration techniques + series)
- PatrickJMT — Integrals / Antiderivatives Playlist
