Consider a two dimensional material. Of course there is no such thing but a flat plate might be modeled as one. The density ρ is a function of position and is defined as follows. Consider a small chunk of area dA located at the point whose Cartesian coordinates are
In other words you integrate the density to get the mass. Now by letting ρ depend on position, you can include the case where the material is not homogeneous. Here is an example.
Example 14.1.6 Let ρ
You need to first draw a picture of the region R. A rough sketch follows.
This region is in two pieces, one having the graph of x = 9y on the bottom and the graph of x = 3y2 on the top and another piece having the graph of x = 9y on the bottom and the graph of
You notice it is not necessary to have a perfect picture, just one which is good enough to figure out what the limits should be. The dividing line between the two cases is x = 3 and this was shown in the picture. Now it is only a matter of evaluating the iterated integrals which in this case is routine and gives 1.