Recall that the procedure for evaluating single variable integrals is to
1. approximate the area under the graph using rectangles; and
2. take the limit of the sum of the area of these rectangles as the number of rectangles approaches
infinity.
Similarly, the volume underneath the function graph for a double integral can be found by
1. dividing the domain R into rectangles;
2. erecting rectangular prisms over these rectangles using the value of the function graph at the
bottom-left vertex of each rectangle as the height;
3. taking the limit of the sum of the prisms’ volumes as the number of rectangles in the domain
aproaches infinity.
Mathematically, this can be stated
\[ \int_R\int f(x,y) dA = \lim_{\|\Delta\| \to 0} \sum_{i=1}^n f(x_i,y_i)\Delta x_i \Delta y_i .\]
