The previous chapter introduced multivariable functions and we applied concepts of differential calculus to these functions. We learned how we can view a function of two variables as a surface in space, and learned how partial derivatives convey information about how the surface is changing in any direction.

In this chapter we apply techniques of integral calculus to multivariable functions. In Chapter 5 we learned how the definite integral of a single variable function gave us “area under the curve.” In this chapter we will see that integration applied to a multivariable function gives us “volume under a surface.” And just as we learned applications of integration beyond finding areas, we will find applications of integration in this chapter beyond finding volume.