Labware - MA35 Multivariable Calculus - Two Variable Calculus



Riemann Integral


A Riemann sum is constructed by dividing a rectangular domain R into sub-rectangles Rij 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 xi, xi+1, yj, yj+1 are the vertices of Rij and cij is a point chosen inside of Rij.

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: \[ \int_R \int f(x,y) dx dy = \lim_{n \to \infty} S_n.\]



  • 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?