Density and Mass

Now, equipped with the double integral, we can consider a lamina with variable density. Suppose the lamina occupies a region D of the xy-plane and its density (in units of mass per unit area) at a point (x,y) in D is given by   , where  is a continuous function on D.

We obtain the mass m of the lamina

Physicists also consider other types of density that can be treated in the same manner. For example, if an electric charge is distributed over a region D and the charge density (in units of charge per unit area) is given by  at a point (x,y) in D, then the total charge Q  is given by

EXAMPLE 1: Charge is distributed over the triangular region D in Figure below so that the charge density at (x,y) is   , measured in coulombs per square meter .Find the total charge.

From Equation 2 and Figure we have

Thus the total charge is