Some Triple Integrals for Mass

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For an object with uniform density, the mass can be calculated as density times volume . For an object with nonuniform density, calculus is necessary. The idea is to partition the object into enough small cubes so that the density of each cube is approximately uniform. Then, the mass of the object, which is the sum of the masses of each of the cubes, can be approximated by the sum of the density times volume for each individual cube (i.e. if the object is partitioned into cubes with density and volume , then the mass of the object is ). The exact mass is the triple integral of the density function.

Contributed by: Laura R. Lynch (June 2014)
Open content licensed under CC BY-NC-SA


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Details

In practice, to find the mass of each individual cube within a partition, choose a representative point within the cube at which to evaluate the density function. Then, approximates the density of the cube and this approximation improves with smaller and smaller cubes. Thus the mass of an individual cube is and the mass of the entire object is . As the number of cubes increases to infinity, this tends to the triple integral: . This Demonstration chooses the midpoint of each cube as the point .

As a special note, snapshot 1 also calculates the volume of the object, because using density function implies that the mass equals the volume.



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