A Full Course on Integration (Multivariable Calculus) - Mastering Line & Multiple Integrals

In this course on integration in multivariable calculus, we explore fundamental and advanced techniques that form the backbone of applied mathematics, physics, and engineering. Starting with exact differentials, we set the foundation for understanding how integrals behave in higher dimensions. Then, we dive into calculating areas enclosed by curves, an essential skill for applications in physics and engineering. One of the most crucial topics covered in this course is line integrals, a concept that extends traditional integration to curves in space. We break it down step by step, starting with what exactly a line integral is before moving to its computation over closed curves and path-dependent and path-independent cases. Using arc length and parametric equations, we develop a deeper understanding of how these integrals work. From there, we move into multiple integrals, learning how to evaluate them efficiently in different coordinate systems. We apply these concepts to real-world problems, such as finding the center of mass, a crucial topic in physics and mechanical engineering. We then extend our knowledge to cylindrical and spherical coordinates, which allow for easier integration in cases of radial symmetry—commonly used in electromagnetism, fluid dynamics, and thermodynamics. By the end of this full course, you'll have a strong grasp of multivariable integration, with practical tools to tackle problems in physics, engineering, and beyond. 00:00:00 – Integration of Exact Differentials 00:07:55 – Area Inside A Closed Curve 00:18:21 – What Exactly Are Line Integrals? 00:30:06 – Line Integrals Around A Closed Curve 00:47:33 – Evaluation of line integrals using arc length and parametric equations 01:05:04 – Line integrals and path independence 01:24:49 – Evaluation of multiple integrals 01:41:03 – Applications of multiple integrals: finding the centre of mass 02:02:57 – Cylindrical coordinates 02:23:15 – Spherical coordinates