Applications of double Integrals: Mass of an object in a two-dimensional space.

 Let's consider a lamina (a thin plate) that occupies the region R of a two-dimensional space. Let (x,y) be a point in the region R surrounded by a small rectangle. The density of the lamina at that point is a function ⍴(x,y). This function is determined by: 

where 𝚫m and 𝚫A represent the mass and the area of the small rectangle.

Let's divide the region R into tiny rectangles of area ΔA. The mass of a tiny rectangle is given by:

Let's add the tiny rectangles together and take the limit of the sum when delta x and delta y approach zero. Since we have two variables, we know by experience that the limit is a double sum of Rieman. This limit represents a double integral that allows us to find the mass off the lamina.

Example

Solution

Let's sketch the region R:

Practice