MAT 1236 Calculus III Section 15.5 Applications of Double Integrals

HW & … WebAssign 15.5 Quiz:15.4, 15.5

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

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

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

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

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.

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

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.

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.

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

Example 2 Find m and