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In this unit, we finish our study of differential equations. This includes a small section on second-order equations. The Briggs textbook includes an electronic-only chapter on these and we include it here. We will only be looking at the first two sections of that chapter (and, even then, only the aspects covered in the lecture videos).

We begin in earnest our study of differential equations. These tremendously useful tools for science and engineering end up being rather pesky in terms of the ways we go about solving them. In this unit, we will be introduced to two types of DEs, as well as a numerical method for approximating solutions.

This unit's material includes three very separate topics. We begin with numerical integration - another of the most important numerical methods we glance at in first year (together with Newton's method, linear approximation, and - soon - Taylor polynomials). The next section covers an interesting detail, looking at integrals that misbehave, either by having an unbounded integrand or an unbounded interval. Either way, the solution is to use limits. Finally, we begin to study differential equations, which are the basis for a tremendous amount of engineering.

This unit, we look at two major methods of integration. Like all of the methods other than IBP, these are actually just subclasses of substitutions, but they are far from obvious and deserve separate treatment. There are a lot of things happening here and plenty of example videos in which to see these details in action. Go through the material carefully and remember that you want to be able to understand when to use each method.

In the rest of this unit's material we begin the study of methods of integration. There is a lot of material in this part of the course and it is important to be able to figure out when to use different methods, so make sure that you are careful in your studies of Chapter 7.

This unit's material is a collection of applications of integrals. The most important section, by far, is 6.7. In that section, we apply integrals to physical problems and start to get a sense of how useful this operation is in interpreting real-world processes.

The remaining lectures in this course give us the framework for applying integration to the real world. Everything that we do here shows up time and again in engineering - Integrals as Net Change allow us to find the total of a changing value; Volumes of Rotation let us find the capacity of conduits and volumes of columns; Arclength lets us look at functions as a part of a building diagram with the knowledge that we can still figure out how much material is needed for construction.

With this unit, we learn about the Fundamental Theorem of Calculus. The great result of the field, it lets us see that integrals and derivatives are (as far as possible) inverse operations. This is a grand result and deserves our careful attention. We also cover the method of substitution, which is the most-used of the ways in which we do integrals (other than, for some people, Wolfram Alpha).

With this unit, we move from derivatives to integrals. We approach these in two different ways: first, we look at what it means to take the inverse of a derivative, and then we try to look at the question of the area under a graph.

A lot of videos in this unit, but three of the four sections have very little in terms of testable material. Only L'Hopital's Rule requires a lot of practice. We are, however, setting the groundwork for some important mathematics. In these sections, we are introducing the idea of numerical methods, setting the stage for integration, and taking the first step toward Taylor series.