This course continues your study of calculus by introducing the notions of series, sequences, and integration. These foundational tools allow us to develop the theory and applications of the second major tool of calculus: the integral. Rather than measure rates of change, the integral provides a means for measuring the accumulation of a quantity over some interval of input values. This notion of accumulation can be applied to different quantities, including money, populations, weight, area, volume, and air pollutants. The concepts in this course apply to many other disciplines outside of traditional mathematics. Through projects, we will apply the tools of this course to analyze and model real world data, and from that analysis give critiques of policy.
Following the pattern as with derivatives, several important methods for calculating accumulation are developed. Our course begins with the study of the deep and significant result of the Fundamental Theorem of Calculus, which develops the relationship between the operations of differentiation and integration. If you are interested in learning more advanced mathematics, this course is the right course for you.
Module 1: Sequences and Series
Calculus is divided into two halves: differentiation and integration. In this module, we introduce the process of integration. First we will see how the definite integral can be used to find the area under the graph of a curve. Then, we will investigate how differentiation and integration are inverses of each other, through the Fundamental Theorem of Calculus. Finally, we will learn about the indefinite integral, and use some strategies for computing integrals.
Module 2: The Definite Integral
In this module, we introduce the notion of Riemann Sums. In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum, named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. This notion of approximating the accumulation of area under a group will lead to the concept of the definite integral, and the many applications that follow.
Module 3: The Fundamental Theorem of Calculus
We now introduce the first major tool of our studies, the Fundamental Theorem of Calculus. This deep theorem links the concept of differentiating a function with the concept of integrating a function. The theorem will consists of two parts, the first of which implies the existence of antiderivatives for continuous functions and the second of which plays a larger role in practical applications. The beauty and practicality of this theorem allows us to avoid numerical integration to compute integrals, thus providing a better numerical accuracy.
Module 4: The Indefinite Integral
In this module, we focus on developing our ability to find antiderivatives, or more generally, families of antiderivatives. In calculus, the general family of antiderivatives is denoted with an indefinite integral, and the process of solving for antiderivatives is called antidifferentiation. This is the opposite of differentiation and completes our knowledge of the two major tools of calculus.
Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
Integration with Calculators and Tables
While the technique of finding antiderivatives is useful, there are some functions that are just too difficult to find antiderivatives for. In cases like these, we want to have a numerical method to approximate the definite integral. In this module, we introduce two techniques for solving complicated integrals: using technology or tables of integrals, as well as estimation techniques. We then apply our knowledge to analyze strategies and decision theory as applied to random events.