Dave4Math / Courses / Applications of the Integral

Applications of the Integral

Learn how to apply the definite integral in science and engineering. In this course, you'll master applications of the integral and become experienced with hyperbolic functions.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Do you have high expectations for your calculus instructor? Why settle for less? Introducing Dave’s online math course: Applications of the Integral. Dave is fantastic, and he really cares about students more than others. Shouldn’t you enjoy the subject while you master it?

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in applications of the integral.
  • Students who are enrolled in calculus 2 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 3.
  • Anyone wanting to learn about areas between curves, various methods for finding volume, finding arc length, understanding work, other applications, and hyperbolic functions.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Applications of the Integral

  • A Real-Life Interpretation
  • The Area Between Two Curves
  • Integrating with Respect to y
  • What Happens When the Curves Intertwine?
  • Solids of Revolution
  • The Disk Method
  • The Washer Method
  • The Method of Cross Sections
  • The Method of Cylindrical Shells
  • Applying the Method of Cylindrical Shells
  • Shells by Revolving About the x-axis
  • Definition of Arc Length
  • Length of a Smooth Curve
  • The Arc Length Function
  • Surfaces of Revolution
  • Work Done by a Constant Force
  • Work Done by a Variable Force
  • Hooke’s Law
  • Moving Nonrigid Matter
  • Work Done by an Expanding Gas
  • Fluid Pressure
  • Force Exerted by a Fluid
  • Measures of Mass
  • Center of Mass of a System on a Line
  • Center of Mass of a System in the Plane
  • Center of Mass of Laminas
  • The Theorem of Pappus
  • The Graphs of the Hyperbolic Functions
  • Hyperbolic Identities
  • Inverse Hyperbolic Functions
  • Derivatives and Integrals of Hyperbolic Functions
  • Derivatives of Inverse Hyperbolic Functions
  • An Application

Course Description

This course is a continuation of the theory of integrals from calculus 1. So we begin this course with several applications of the integral. First up is the area between curves. We motivate and prove theorems that give us the flexibility of finding areas from different viewpoints. In doing so, we work through several examples that demonstrate each strategy.

After finding areas, we continue with finding volumes. We begin by explaining solids of revolution and then walk through (in detail) several methods for find volumes, including using disks, washers, cross-sections, and cylindrical shells.

We then turn to arc length and its generalization: areas of surfaces of revolution. We motivate these concepts and work through several examples and applications. In detail, we examine work, fluid pressure and force, and moments and center of mass.

At the end of this course, we study hyperbolic functions and some applications. We give a thorough treatment by defining these functions, examining their graphs, exploring fundamental identities, proving derivative formulas and derivative formulas of their inverses.

Recommended Prerequisites for Applications of the Integral

I recommend the prerequisite course Theory of Integrals. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 2 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.