Integrating Functions of Several Variables

In single-variable calculus, recall that we used the classic calculus approach to define the definite integral as the area under a graph. Specifically, we approximated the area under the graph of a positive function \(f\) on an interval \([a,b]\) by adding areas of rectangles whose heights are determined by the curve. We then broke the interval \([a,b]\) into smaller subintervals, constructing rectangles on each of these smaller intervals to approximate the region under the curve on that subinterval, then summing the areas of these rectangles to approximate the area under the curve. We defined the definite integral of \(f\) using the limit of this Riemann sum as the size of all of the subintervals goes to zero.In this chapter we will expand our idea of integration and accumulation to apply to scalar valued, multivariable function.