Calculus Revisited: Multivariable Calculus

Part V: Multiple Integration

In dealing with functions of a single variable, a definite integral can be viewed as representing the mass of a thin rod. In this interpretation we are integrating along a straight line segment. On the other hand we may view the same integral as denoting the area of the region bounded by a given curve y = f(x), the x- axis and two lines that are parallel to the y-axis. However since the answer is the same in either case no real distinction is made between these two interpretations. However in the case of functions of more than a single variable the two interpretations are quite different. For example, if we are dealing with a function of two real variables, the curve z = f(x,y) defines a region in the plane and it makes sense to talk about what is happening inside the region and what is happening on the boundary of the region. How we distinguish between these two interpretations is the subject matter of Part V.

Lecture 1: Double Multiple Sums

Lecture 2: The Fundamental Theorem

Lecture 3: Multiple Integration and the Jacobian

Lecture 4: Introduction to Line Integrals

Lecture 5: Green's Theorem