Calculus Revisited: Single Variable Calculus

Part IV: The Definite Integral

Differential calculus introduces limits to extend the concept of average rates of change to instantaneous rates of change. The subject known as integral calculus introduces limits to extend the technique for finding the area of rectilinear regions (that is, regions that have straight line borders such as triangles and rectangles) to finding areas of non-rectilinear regions (e.g., circles). Differential calculus and integral calculus are related through the definite integral. The definite integral represents the area of a non-rectilinear region and the remarkable thing is that one can use differential calculus to evaluate the definite integral. How this is done is the topic of this part of our course, which culminates with a discussion of what are called the fundamental theorems of calculus.

Lecture 1: The Definite Integral

Lecture 2: Marriage of Differential and Integral Calculus

Lecture 3: Three-Dimensional Area

Lecture 4: One-Dimensional Area