INSTRUCTOR: Consider the function z equal to f of x comma y, equal to 3x squared minus y, over the rectangular region R equals 0 comma 2 by 0 comma 2. Divide R into the same 4 squares with m equals n equals 2, and choose the sample points as the upper left corner point of each square. That is the points 0 comma 1. The point 1 comma 1. The point 0 comma 2, and the point 1 comma 2.

To approximate the assigned volume of the solid S that lies above R and under the graph of f. Well, first we need to find what delta A is. By definition, delta A is delta x multiplied by delta y. If you notice how these points are laid out, we have change in x of 1, and a change in y of 1. So that delta A is just 1 multiplied by 1.

Now, the way that our sum is set up to approximate our values, our volume. V will be equal to f of the point 0, 1. 0 comma 1 multiplied by delta A, which is 1, plus f of 1 comma 1, multiplied by delta A, which is 1, plus f of 0 comma 2 multiplied by 1, plus f of 1 comma 2 multiplied by 1 as well. Now evaluating our function at each of these upper left corner points, this becomes negative 1 plus 2 minus 2 plus 1, which gives us a total signed volume of 0.