# 14.8: Infinitesimal Calculus for integration

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This chapter discussing the calculus of integration in a very brief manner. Proofs are limited and applications are more emphasized here. Most of the sections here are from other Libretexts courses/books rather than the main authors of this course. Tables, applications, and tidbits are herein but nothing more and as such this does not replace a calculus course where you will learn most of the theory properly. Here we are really only concerned with how integrals are used in engineering and science. While you should try and understand the basics in this chapter, it is not necessary to understand the full scope presented here as you will have that opportunity in your calculus class proper.

As we stated previously, integrals can be defined by a Riemann sum.

$F(x) = \int_a^b {f(x)dx} = \lim_{||\Delta x|| \to 0} \sum_{i=1}^n f(x_i) {\Delta x}$

This is not a perfect definition, but is sufficient for our proposes here. A true integral is actually an antiderivative, so the derivative is the place to look for the idea on an integral. We will leave the complexities of integration for your calculus class and in the preceding sections give a table of integrals and some application.

Finally, the following Wolfram Demonstration exemplifies the Riemann sum. You can change the number of boxes (n) to reduce your $$\Delta x$$ but cannot, obviously, have $$||\Delta x||$$ approach zero. With this you can imagine it though.

 Example of Riemann Sums using a Wolfram Demonstration Project. The controls are useable for you to try to reduces the $$\Delta x$$. Phil Ramsden, "Riemann Sums: A Simple Illustration", http://demonstrations.wolfram.com/RiemannSumsASimpleIllustration/,Wolfram Demonstrations Project, Published: March 7 2011

• 14.8.1: Table of Integrals
Table of integrals.
• 14.8.2: Substitution
In this section we examine a technique, called integration by substitution, to help us find antiderivatives. Specifically, this method helps us find antiderivatives when the integrand is the result of a chain-rule derivative.
• 14.8.3: Applications of Integration
In this section, we use definite integrals to calculate the force exerted on the dam when the reservoir is full and we examine how changing water levels affect that force. Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world.

14.8: Infinitesimal Calculus for integration is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.