Labware - MA35 Multivariable Calculus

A Riemann sum is constructed by dividing a rectangular domain R into sub-rectangles R_ij and multiplying their area by a height funtion f. This is commonly denoted \[ S_n = \sum_{i,j=0}^{n-1} f(c_{ij}) (x_{i+1}-x_i) (y_{j+1}-y_j), \] where x_i, x_i+1, y_j, y_j+1 are the vertices of R_ij and c_ij is a point chosen inside of R_ij.

The method for summing the volume under a function graph described in the previous section used a height function which took the lowest point on the function graph above the sub-rectangle, called a lower sum. Similarly, an upper sum can be used by using a height function which takes the highest point on the function graph above the sub-rectangle.

A Riemann integral is obtained by letting the number of divisions in a Riemann sum go to infinity: \[ \int_R \int f(x,y) dx dy = \lim_{n \to \infty} S_n.\]

Demos

Upper Rectangular Prisms

This demo shows the Upper Sum for the function graph of f(x,y)=-x² - 0.1y² + .2, whose Upper Sum was already shown in part section 2.4.1.

Difference of Rectangular Prisms

This last demo effectively summarizes the connection between Upper and Lower Sums. The Upper Sum gives an upper bound on the value for the volume under the function graph, and the Lower Sum gives a lower bound. Increasing the "res" variable, you can see that the difference between the Upper and Lower Sum grows smaller and smaller. This demonstrates that the Upper and Lower Prisms converge to the same volume, and thus the Riemann integral of the volume under a function graph is defined.

Exercises

1. In single variable calculus, midpoint Riemann integrals are offered as an alternative to lower and upper sums. Is there a similar alternative to lower and upper sums for integrals over two variables?

2. Another alternative to lower and upper sums in single variable calculus is the trapezoidal approximation. Find an analogous alternative to the rectangular prisms used in this lab and in lab 2.4.1, and describe this alternative explicitly.

3. What other alternatives to lower and upper sums can you come up with?