Calculus through Data & Modelling: Series and Integration

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.

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.

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.

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.