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Applications of Double Integrals: Mass and Center of Mass of Thin Plates in Calculus

This resource covers key applications of double integrals, focusing on the mass and center of mass of thin plates (lamina) with uniform and non-uniform density. It explains how to calculate the mass of a thin plate with a given density function over a defined region D and discusses methods for determining the center of mass, emphasizing the importance of defining the region correctly. The examples provided illustrate the concepts using both uniform and variable density distributions, making the principles accessible and applicable to various real-world scenarios.

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Applications of Double Integrals: Mass and Center of Mass of Thin Plates in Calculus

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  1. MAT 1236Calculus III Section 15.5 Applications of Double Integrals http://myhome.spu.edu/lauw

  2. HW & … • WebAssign 15.5 • Quiz:15.4, 15.5

  3. Preview • Mass of a thin plate (lamina) • Center of mass (formula only)

  4. Thin Plate - Lamina • A thin plate with uniform density, occupies a region D. • The mass of the thin plate is given by

  5. Thin Plate with Non-Uniform Density • A thin plate with density (x,y), occupies a region D.

  6. Thin Plate with Non-Uniform Density • A thin plate with density (x,y), occupies a region D.

  7. Remarks • Other type of density can be treated the same. • If (x,y) is the electric charge density, then is the total charge in D.

  8. Example 1 • Electric charge is distributed over the disk so that the charge density at is • Find the total charge on the disk.

  9. Expectations • Be sure to define the region D before you use it. • In the case that the descriptions are in rectangular coordinates, you need to give two set descriptions for D.

  10. Center of Mass • Physical significance: the lamina behaves as if its entire mass is concentrated at its center of mass. • The lamina balances horizontally when supported at its center of mass.

  11. Center of Mass A thin plate with density (x,y), occupies a region D.

  12. Example 2 Find m and

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