Calculus 2, part 1 of 2: Integrals with applications

Calculus 2: Mastering Integrals & Applications

This comprehensive course delves into the world of integral calculus, building upon your single-variable calculus foundation. You'll gain a deep understanding of integrals and their powerful applications in solving real-world problems. This first part of the course (of two) provides a solid foundation for further study.

Course Highlights:

• Antiderivatives & Integration Techniques: Learn the fundamental concept of the antiderivative (also known as the primitive function) and master essential integration techniques.
• Integration by Parts: Explore the integration by parts method, a crucial technique for evaluating integrals involving products of functions. We cover standard applications, including those with sine/cosine and polynomials, exponential functions, and even more advanced examples involving arctangent and logarithms.
• Change of Variables: Discover how to effectively use change of variables (the "reverse chain rule") to simplify complex integrals.
• Partial Fraction Decomposition: Become proficient in integrating rational functions using partial fraction decomposition.
• Trigonometric Integrals: Learn various methods for evaluating integrals containing trigonometric functions, including trigonometric identities, the tangent half-angle substitution, and other clever techniques.
• Advanced Substitution Techniques: Explore Euler substitutions and triangle substitutions (trigonometric substitutions) for tackling challenging integrals.
• Riemann Integrals: Understand the formal definition of Riemann integrals (definite integrals) and their connection to area under a curve. We'll cover partitions, Riemann sums, integrable functions, and prove key theorems like the Mean Value Theorem for Integrals.
• Integration by Inspection: Develop the ability to quickly evaluate integrals of common functions, including those related to geometric shapes and periodic functions.
• Fundamental Theorem of Calculus: Master the Fundamental Theorem of Calculus, a cornerstone of calculus. Learn how to use it for evaluating integrals, computing limits, and finding derivatives of functions defined by integrals.
• Area Between Curves: Calculate the area enclosed between two or more continuous curves.
• Arc Length: Determine the arc length of a curve.
• Rotational Volume: Compute volumes generated by rotating a region around an axis.
• Surface Area: Calculate the surface area of a solid generated by rotating a curve around an axis.
• Improper Integrals: Evaluate improper integrals of both the first and second kinds, dealing with infinite limits of integration or unbounded integrands.
• Convergence Criteria: Use comparison criteria to determine the convergence of improper integrals.

Problem Solving & Applications:

Throughout the course, you'll have ample opportunities to practice these techniques through numerous solved problems. You'll also receive a brief introduction to initial value problems, a topic covered in more depth in future Ordinary Differential Equations (ODE) courses.

Important Note: The specific topics covered on your final exam may vary depending on your institution. Always consult with your professor to determine the relevant material.

Resources:

A detailed list of all videos and solved problems is available in the resource file: “001 List_of_all_Videos_and_Problems_Calculus_2_p1.pdf” (accessible under Video 1 – "Introduction to the course"). This information is also presented within Video 1.