
MTH243 (Calculus
for Functions of Several Variables)
MATHEMATICA. Chapter 16:
Integrating Functions of Several Variables
Vladimir A. Dobrushkin,Lippitt
Hall 202C, 8745095,dobrush@uri.edu
In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI.
The Mathematica projects are created to help you learn new concepts. Mathematica is very useful in visualizing
graphs and surfaces in three dimensions. Matlab (commercial software) is also available at engineeering labs. Its
free version is called Octave. A student can also use free CASs: SymPy (based on Python), Maxima, or Sage.

Section 16.1. The Definite Integral
The definite integral of a function \( f (x, y) \) over a rectangular domain
\( R: \ a\le x \le b, \ c \le y \le d \) is
\[
\int_R f\, {\text d} A = \int_R \, {\text d}x\,{\text d} y \, f(x,y) = \lim_{\Delta x , \Delta y \mapsto 0} \, f\left( u_{ij}, v_{ij} \right) \Delta x \, \Delta y
\]
down = {0, 0} + # {1.5, 1} & /@ Range[0, 10];
up = {0, 1} + # {1.5, 1} & /@ Range[0, 9];
ListLinePlot[Riffle[down, up], Filling > Axis]
Plot[Floor[x], {x, 0, 15}, Filling > Axis]
Another graph
raggedTriangle[pmin_?VectorQ, pmax_?VectorQ, p_: 5] :=
Module[{d = pmax  pmin, iv}, iv = Subdivide[#, 100/p] & /@ d;
Polygon[
Append[TranslationTransform[pmin] /@
Most[Riffle[Transpose[iv],
Transpose[MapAt[RotateLeft, iv, 2]]]], {First[pmax],
Last[pmin]}]]]
Graphics[{{LightGreen, Rectangle[{1, 2}, {5, 3}]}, {Orange,
raggedTriangle[{1, 2}, {5, 3}, 10]}}]
Example 2. Let R be the rectangle
\( 0 \le x \le 1 \) and \( 0 \le y \le 1 . \)
Use Riemann sums to make upper and lower estimates of the volume of the region above R and under the graph of
\( z = e^{(x^2 + y^2 )} . \)
Example 3 . Let \( f (x,y) = x^2 y \)
and let R be the rectangle
\( 0 \le x \le 1 , \ 0 \le y \le 1 . \)
Show that the difference between upper and lower Riemann sums for f on R converges to 0, as
\( \Delta x \ \mbox{ and } \ \Delta y \) approach 0.
 