Introduction to Integration

Introduction to Integration

Previous chapters dealt with Differential Calculus. They started with the "simple" geometrical idea of the slope of a tangent line to a curve, developed it into a combination of theory about derivatives and their properties, techniques for calculating derivatives, and applications of derivatives. This chapter begins the development of Integral Calculus and starts with the "simple" geometric idea of area. This idea will be developed into another combination of theory, techniques, and applications.

One of the most important results in mathematics, The Fundamental Theorem of Calculus, appears in this chapter. It unifies the differential and integral calculus into a single grand structure. Historically, this unification marked the beginning of modern mathematics, and it provided important tools for the growth and development of the sciences.

The chapter begins with a look at area, some geometric properties of areas, and some applications. First we will see ways of approximating the areas of regions such as tree leaves that are bounded by curved edges and the areas of regions bounded by graphs of functions. Then we will find ways to calculate the areas of some of these regions exactly. Finally, we will explore more of the rich variety of uses of "areas".

The primary purpose of this introductory section is to help develop your intuition about areas and your ability to reason using geometric arguments about area. This type of reasoning will appear often in the rest of this book and is very helpful for applying the ideas of calculus.

Source: Dale Hoffman,
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