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and aerospace engineers involves the theory of beams. A fundamental concept in beam theory is the idea of a centroid, or more generally, the center of gravity. Given a strangely-shaped region cut out of stiff cardboard, students can readily understand the idea of center of mass: it's the point where the region would exactly balance on the tip of a pencil. A more difficult problem is how to compute the location of that point, especially if the material is not symmetric and is not of uniform density.

As soon as the multivariable calculus students are introduced to double integrals, they are immediately asked to think about the geometric ideas associated with numerically approximating the area of a planar region. They use Maple to help them explore the idea of superimposing a grid on a region and adding up the area of all squares that are "inside" the region. The region we give them is a model of the state of Minnesota (see Figure 1); the location of the state boundary was derived from a cartographic database. We call this the "state of Minnesota lab".

### Figure 1:(Left) A model of the state of Minnesota with a superimposed 10 x 10 grid. If you know that scale factors of the model, can you bound the area of the state? (Right) A model of a population distribution for the state of Minnesota. Students are asked to find the demographic center of the state.

The students quickly ask a fundamental question: "if a square is only partially contained in a region, do we include its area in the final sum?" This leads to a discussion of upper bounds, lower bounds, and error estimates. It is not long before a second fundamental question is proposed: "If we want to know the area better, can't we just use a finer grid?" This leads to discussions of convergence, Riemann sums, and numerical methods for approximating double integrals. Furthermore, the students recognize these somewhat abstract mathematical ideas as being useful and natural, rather than thinking of them as artificial constructs that are unrelated to engineering. Eventually each group agrees on a number to assign "the" area of the model. For homework they are asked to look up the true area of the state and to compare their estimate with the official value; most estimates are within 5%.

Once the students have learned and extended some of these ideas, they are asked to compute the centroid of the state, and then to consider what happens to the "center" of Minnesota if we do not assume a uniform density distribution. Specifically, we create a model that takes into account a real geological feature: the Great Iron Range of northern Minnesota. The model represents the Iron Range by assuming a density distribution which is highest in a roughly east-west swath across the upper portion of the state. The students discover that if we assume such a density distribution, then the state's "geological" center of mass is further north-east than if we assume a uniform distribution. The students then look at a model of a population distribution in which the bulk of the state's population lies in the eastern portion of the state, and in which the state's population decreases from south to north. They conclude that the demographic center of this model is just northwest of the Twin Cities.

The students are now equipped to take a density function for any distribution of objects (people, soybeans, income, voters, etc.) and find the center of mass for that distribution, but they also learn more from this lab than just how to find a center of mass. They learn to question the accuracy of a model and to interpret a model's predictions. They learn that some problems have no closed-form analytic solutions. They learn both theoretical and numerical techniques of integration, and why each is important. Finally, they learn these concepts better with by using technological tools, because of the rich exploration that technology permits.

Next: The Geometry of Fluid Flow
Up: Abstract
Prev: Multivariable Mathematics