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A Complete Guide to Integral Calculus (Advanced Calculus)

Name: A Complete Guide to Integral Calculus (Advanced Calculus)
Rating: 5.0 (2 reviews)

Mastering Integral Calculus: Jacobians, Gamma Functions, and Surface Integrals

Created byGina Chou

Last updated 1/2025

English

What you'll learn

Understand and apply the concept of Jacobians in polar coordinates to transform functions of two variables.
Evaluate double integrals by changing variables, simplifying computations for complex regions.
Apply double integrals to solve real-world problems, including area, mass, and center of mass calculations in polar coordinates.
Analyze and evaluate integrals related to the Gamma function and its properties.
Comprehend the properties and applications of the Gamma function and use it in integral calculus problems.
Apply the Laplace transform to solve integrals and understand its importance in mathematical modeling.
Understand Jacobians in cylindrical and spherical coordinates and apply them for coordinate transformations in three-dimensional problems.
Perform triple integrals by changing variables to cylindrical or spherical coordinates to simplify integrals.
Apply triple integrals to real-world contexts, such as calculating volumes, masses, and centers of mass for 3D objects in cylindrical or spherical coordinates.
Calculate surface area using integration techniques for surfaces defined parametrically or in coordinate systems.
Evaluate surface integrals in cartesian, cylindrical, and spherical coordinates.
Apply surface integrals to solve problems in physics and engineering, such as flux and surface area calculations in various coordinate systems.

Course content

6 sections • 43 lectures • 4h 45m total length

Overview1:38
Welcome and How It Works0:30
Tips to Maximize Your Learning0:34

Downloadable Notes0:03
Overview of Section 20:23
Element of Area10:39
Jacobian11:14
derive the area element in generalized coordinates using the jacobian of x and y with respect to u and v, and illustrate with cartesian and polar coordinates.
Transformation of Coordinates in Double Integrals7:19
Examples: Transformation of Coordinates in Double Integrals15:34
Applications of Double Integrals3:10

Downloadable Notes0:03
Overview of Section 30:23
Gamma Function6:25
Explore the gamma function defined by the integral from zero to infinity of e^{-x} x^{n-1} dx, with recursion gamma(n+1)=n gamma(n) and gamma(1)=1, yielding gamma(n+1)=n! for positive integers.
To Obtain Gamma(1/2)8:33
Compute gamma of one half by transforming the gamma integral with substitutions and polar coordinates to obtain gamma(1/2)=sqrt(pi), and apply the recursion gamma(n+1)=n gamma(n) to reach gamma(3/2) and beyond.
Evaluation of Some Important Integrals using the Gamma Function9:40
Derive an equation to evaluate integrals involving products of sine and cosine using the gamma function, and relate the integral to gamma(n) and gamma(m) via polar coordinates.
Examples: Evaluating Integrals using the Gamma Function11:57
Study how to evaluate integrals with the gamma function through three examples, handling cosine and sine powers, quadrant rules, and the zero result before previewing the Laplace transform.
Laplace Transform16:00

Downloadable Notes0:03
Overview of Section 40:23
Triple Integrals in Cylindrical Coordinates9:37
Evaluate triple integrals in cylindrical coordinates to compute the hurricane's total kinetic energy from density delta0 e^{-z/h} and theta-directed velocity, identifying max velocity at r=a, z=0.
Transformation of Coordinates in Triple Integrals11:01
Learn to transform coordinates in triple integrals by expressing x, y, z as functions of u, v, w using the Jacobian determinant and explore cylindrical coordinates with r, theta, z.
Example 1: Triple Integrals in Cylindrical Coordinates11:23
Example 2: Triple Integrals in Cylindrical Coordinates12:55
Example 3: Triple Integrals in Cylindrical Coordinates8:56
Compute the mass of the smaller part of a sphere cut by z = 1/2 with density, via a triple integral in cylindrical coordinates, switching the order to yield pi/4.
Integral Calculus Part 1
Spherical Coordinates7:31
Example 1: Triple Integrals in Spherical Coordinates11:04
Example 2: Triple Integrals in Spherical Coordinates10:35
Example 3: Triple Integrals in Spherical Coordinates4:43
Example 4: Triple Integrals in Spherical Coordinates7:16
Learn to compute the moment of inertia and radius of gyration for a solid homogeneous sphere of constant density using triple integrals in spherical coordinates.
Revisit Lecture 24 using Spherical Coordinates6:41
Revisit Lecture 22 using Spherical Coordinates9:47

Downloadable Notes0:03
Overview of Section 50:23
Examples of Surface Area10:34
Surface Integrals in Cartesians3:58
Example 1: Surface Integrals in Cartesians6:53
Example 2: Surface Integrals in Cartesians10:17
Example 3: Surface Integrals in Cartesians6:57
Surface Integrals in Spherical Coordinates4:18
Compute surface areas in spherical coordinates by applying the cross product magnitude of r_phi and r_theta, yielding the integrand a^2 sin phi and setting bounds for phi and theta.
Example 1: Surface Integrals in Spherical Coordinates16:40
Example 2: Surface Integrals in Spherical Coordinates8:45
Integral Calculus Part 2

Requirements

Proficiency in Single-Variable Calculus (Calculus 1 and 2): A solid understanding of differentiation and integration for single-variable functions, including the Fundamental Theorem of Calculus, techniques of integration, and applications of single-variable integrals.
Introductory Multivariable Calculus Knowledge (Calculus 3): Familiarity with partial derivatives, double and triple integrals, and basic coordinate transformations (e.g., Cartesian to polar).
Basic Linear Algebra Skills: Understanding of vectors and matrices, which is helpful for transformations and working with Jacobians.

Description

How This Course Works

This course, A Complete Guide to Integral Calculus (Advanced Calculus), builds upon foundational calculus to dive deeply into the behavior and applications of functions in multiple dimensions. Focusing on integral calculus, it also provides an introduction to key vector calculus concepts that follow in the complete series, including major theorems like Green’s, Stokes’, and the Divergence Theorem. This course is essential for students in mathematics, physics, engineering, and other fields who want a practical and theoretical understanding of multivariable calculus tools for real-world problem-solving.

Who Should Take This Course?

This course is ideal for university students currently enrolled in Advanced Calculus, or those who have completed Calculus III and Linear Algebra. It’s also designed for anyone eager to explore multivariable calculus applications more deeply, especially in fields where calculus techniques are indispensable.

Course Overview

This course includes Integral Calculus, the first part of the complete Advanced Calculus series. You’ll have access to lecture videos, whiteboard notes, and problem sets with solutions. Topics covered here include:

Integral Calculus (Sections Included in This Course)

Two-Variable Functions: Explore Jacobians in polar coordinates, variable changes in double integrals, and applications of double integrals.
Gamma Function and Laplace Transform: Understand a key integral related to the Gamma function, the Gamma function itself, and the Laplace transform.
Three-Variable Functions: Learn about Jacobians in cylindrical and spherical coordinates, transformations in triple integrals, and applications of triple integrals.
Surface Area and Surface Integrals: Calculate surface areas and evaluate surface integrals in Cartesian, cylindrical, and spherical coordinates.

Vector Calculus, Integral Theorems, and Differential Calculus of Several Variables (Available in the Complete Course)

Vector and Scalar Fields
Line Integrals
Flux, Circulation, and Vector Operators
Integral Theorems
Introduction to Partial Differential Equations
Topics in Differential Calculus of Several Variables

Course Content

Videos: Each topic is introduced and explained thoroughly, with step-by-step examples, making complex problems manageable.
Notes: Downloadable lecture notes for each section to allow for offline review and reinforcement.
Assignments: Practice problem sets with solutions so you can check your work after attempting each problem yourself.

Highlights of What’s Included

Lifetime access to A Complete Guide to Integral Calculus (Advanced Calculus).
Downloadable lecture videos and notes for anytime review.
Two comprehensive problem sets with solutions to reinforce learning.
An instructor committed to your success every step of the way.

See you inside the course!

– Gina Chou :)

Who this course is for:

Undergraduate and graduate students in mathematics, physics, engineering, and related fields.
Seek a deeper understanding of integral calculus concepts and techniques for multi-variable functions.
Wish to develop skills in using advanced calculus tools like Jacobians, Gamma functions, and Laplace transforms to simplify and solve complex integrals.
Aim to apply integral calculus to real-world problems in areas such as fluid dynamics, electromagnetism, and probability theory.

A Complete Guide to Integral Calculus (Advanced Calculus)

What you'll learn

Explore related topics

Course content

Introduction3 lectures • 3min

Integral Calculus of Two-Variable Functions7 lectures • 48min

Gamma Function and Laplace Transform7 lectures • 53min

Integral Calculus of Three-Variable Functions14 lectures • 1hr 52min

Surface Area and Surface Integrals10 lectures • 1hr 9min

Conclusion2 lectures • 1min

Requirements

Description

Who this course is for: