Unit 3: Integration in the Multivariable Context

This unit covers the theory, techniques, and applications of integration in the multivariable context. As you'll discover, integrating multivariable functions like f(x,y)=x^2+y^2 requires multiple integrals. That may sound scary, but we'll develop the content slowly enough that it will seem like a natural extension of what we did in single-variable integration. We'll start by studying double integrals, which we'll first think of as simply "integrating twice." Then we'll move on to understanding how to integrating two-variable functions over more complicated regions that would benefit from thinking more carefully about the region of integration. Later, we'll consider triple integrals, which are needed when integrating a function of three variables. In between, we'll learn how to change coordinates systems and integrate in cylindrical and polar coordinates.

We'll being our study of multiple integration in Lesson 10. There we'll study how to integrate functions defined over the simplest two- and three-dimensional regions: rectangles and rectanguloids (i.e., boxes).

In Lesson 11 we'll generalize our work to handle functions defined over more general 2D and 3D regions.

In Lesson 12 we'll explore alternative coordinate systems for evaluating double and triple integrals. We'll focus on two in the lesson itself -- polar and spherical coordinates -- and introduce another -- cylindrical coordinates -- in the practice problems.

Thus far in the course we've spent all our time learning about three-dimensional space, vectors, functions, and the limits and derivatives of multivariable functions. We've yet to study integration, one of the principal branches of calculus. This unit fills that gap. It also sets us up for the next unit, which combines the vector content we've learned with the derivatives and integration content.