Volume Under a Function Graph
(Page: 1
 2 ) Text Just as in the Cartesian case, the definite integral of a function of two variables in polar coordinates represents the volume underneath its
threedimensional function graph.
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
bottomleft 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.
Demos
Approximation by Rectangular Prisms
 
This demo shows the volume under the graph of a function f(x(r, t), y(r, t)) (where x and y are defined in terms of polar coordinates r and t) for a given domain, which you may change. You can also change the number of steps to get a more accurate approximation.

