3D Riemann Sum on Rectangles
When you want to calculate the area taken up by a multivariable function in a rectangle defined by the set of points which can also be written as the rectangle drawn between the points and , we can split up the rectangle into smaller rectangles of area . After taking a sample point from each of those rectangles, we can add up the values of those sample points to calculate an estimate of the space used. Obviously, as decreases, the estimate gets more accurate.
The Riemann sum of such a rectangle is therefore defined by:
$$ \sum_{i=0}^m\sum_{j=0}^n f(x_{ij}^, y_{ij}^) \Delta x \Delta y $$