Calculus through Data & Modelling: Series and Integration

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