Section 1 - Introduction to Calculus and the History of Integration

In this section, we introduce the integration part of Calculus 2 and explore the concept of integration, its historical background, and the major contributions that have evolved from ancient times to the modern era. In particular, we cover everything from classical approaches such as the method of exhaustion to the formalization of calculus by Newton and Leibniz.

Section 2 - Definition of Riemann Sums and Definite Integrals

Understand the basic concept of definite integrals through Riemann sums, and learn the definition and various properties of definite integrals. Through this, you will build a foundation for calculating the area of regions enclosed by curves.

Section 3 - Indefinite Integrals and Basic Integration Techniques

Learn the concept of the indefinite integral, which is the inverse operation of differentiation, and master key techniques for calculating indefinite integrals for various functions. In particular, integration by parts and integration by substitution are covered in depth.

Section 4 - Fundamental Theorem of Calculus and Advanced Integration Techniques

You will learn the Fundamental Theorem of Calculus by Leibniz, which is the core of calculus, and study advanced techniques for calculating the integrals of various functions, including logarithmic functions, trigonometric substitution, and rational functions.

Section 5 - Applications of Integration: Volume, Arc Length, and Surface Area

Explore how to apply the concept of integration to real-world problems. Utilize integration to solve various geometric problems, such as calculating the volume of solids of revolution, measuring the arc length of curves, and calculating the surface area of solids of revolution.

Section 6 - Improper Integrals and Numerical Integration

We will cover improper integrals where the interval of integration is infinite or the integrand has singularities, and learn numerical integration methods to approximate the values of complex functions, along with implementation using Python code.

Point 1. Starting strong with the history of calculus!

From the volume calculations of ancient Egypt to the area determinations of ancient Greece, you will learn the fascinating history and principles of integration in depth. Solidify your understanding of the fundamental concepts of integration through classical methods such as the 'method of exhaustion.'

Point 2. Definite Integrals: Easy Understanding and Application!

Gain a clear understanding of the definition of definite integrals through Riemann sums and Darboux definitions, and learn how to solve complex problems using Leibniz's Fundamental Theorem of Calculus. Apply these concepts directly to calculating area, volume, and arc length.

Point 3. Mastering Various Integration Techniques!

Systematically learn various integration techniques, including integration by parts, substitution, and trigonometric substitution. We will also cover the integration of rational functions, building your ability to integrate functions of any form.

Point 4. Numerical Integration with Python!

You will learn numerical integration methods useful for handling real-world data along with Python coding. Beyond theoretical learning, you can gain practical experience in understanding and applying the concepts of numerical integration by writing the code yourself.