Integration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.
Imagine you have a car driving along a road, and its speed changes over time. At some moments, it’s going faster; at other moments, it’s slower. If you want to find out how far the car travelled over a period of time, you can’t just multiply “speed × time” because the speed isn’t constant.
Instead, you break the trip into tiny pieces of time (like seconds) where the speed doesn’t change much. For each tiny piece of time, you calculate how far the car went at that speed, then add up all those tiny distances.
This section covers key integration concepts, methods, and applications, including the Fundamental Theorem of Calculus, integration techniques, and how to find areas, volumes, and other geometric properties.
- Introduction to Integration
- Antiderivative
- Fundamental Theorem of Calculus
- Functions defined by Integrals
This section builds the idea of integration from sums, showing how definite integrals arise as limits of Riemann sums and how they are evaluated.
- Riemann Sum
- Riemann Sums in Summation Notation
- Definite Integral as the Limit of a Riemann Sum
- Definite Integrals
- Properties of Definite Integrals
- Evaluation of Definite Integrals
Here, you’ll learn about different forms of integrals and when to use each type to represent and solve accumulation problems.
- Indefinite Integrals
- Improper Integrals
This section introduces powerful techniques and standard formulas that help simplify and evaluate a wide variety of integrals.
- Integration Formulas
- Methods of Integration
- Integration by Substitution
- Integration by Trigonometric Subtitution
- Integration by Parts
- Integration by Partial Fraction
This section introduces powerful techniques and standard formulas that help simplify and evaluate a wide variety of integrals.
- Application of Integration (Overview)
- Area Under a Curve
- Area Between Curves
- Area Between Polar Curves
- Volume of Solids of Revolution
- Arc Length of Curves
- Surface Area of Revolution
This section extends integration to functions of several variables and vector fields, enabling the study of higher-dimensional problems.
- Double Integration
- Triple Integral
- Line Integral
- Surface Integral
This section provides quizzes and practice questions on integration, covering key topics like basic integration, applications, and integration by substitution.
- Integration - Quiz
- Applications of Integration - Quiz
- Practice Question on Integration by Substitution