MAT 1236 Calculus III

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