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Modeling the real world: The applications of the definite integral are numerous and varied with applications ranging from physics, to engineering, to satellite orbits. Thus, Riemann sums, as an approximation to the definite integral, also have many real-world applications. In the following we include applications to export data (adapted from Calculus Applied to the Real World by Waner and Costenoble) and population growth.

From figures released by the U.S. Agriculture Department, the value of U.S. pork exports from 1985 through 1993 can be approximated by the equation

Where q represents annual exports (in millions of dollars) and t the time in years, with t = 0 corresponding to January 1, 1985.

The chart below represents an estimation of data that was used to determine the function above. We can use a Riemann process to approximate the total U.S. exports of pork over the given period by summing the individual figures. Note that in this summation we would be adding the areas of rectangles in the figure (since each rectangle has width 1 unit).

If we add up the areas of the rectangles in the chart, we obtain

(60 + 80 + 125 + 250 + 330 + 320 + 335 + 450 + 460)(1)

= $2,410 million.

If we now calculate the left- and right-hand sums of q(t) with n = 9, the number of subdivisions shown in the chart; rounding off to the nearest $1 million, we get

Their average is the trapezoidal sum = $2,294.5 million.

Note that the Riemann sums are close to the true total, so that the model seems reasonable. One cannot expect the actual pork export figures to coincide with the abstract mathematical function. But the model does allow one to predict future exports.

* Problem from A Companion to Calculus by Ebersole, Schattschneider, Sevilla and Somers, Brooks/Cole, 1995. 

Suppose the population growth rate at a time t of a certain rural locale is given by

where t represents years from 1960. (For example, p(1)=206 tells us that at year one the rate of growth is about 206 people per year.) We now use a Riemann sum to estimate the population growth from 1960 to 1975 using five equal subintervals of time and using the left endpoint of each subinterval to estimate the population growth rate for the subinterval.

Solution: The interval from 1960 to 1975 corresponds to 0 £ t £ 15. If we divide this interval into five equal

subintervals, each has width = 3; therefore,

the endpoints of the subintervals are 0, 3, 6, 9, 12, and 15.

The Riemann sum to estimate the population growth is as follows:

= 3 [ 200 + (200 + 6Ö 3) + (200+6Ö 6) + (200+6Ö 9) + (200+6Ö 12) ]

@ 3192.

USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil