Different uses of the double integral?

[hivesaeed4]

Could somone tell me how is it that the double integral could be used for both calculating the area as well as the volume? And please explain that how does the triple integral, which is used to find the volume as well, fits in the picture and how is it different from the double integral we use to calculate the volume?

 

[mathman]

It depends on what the variables of integration represent. You need to describe the situations you are talking about. Usually a single integral gives area, a double integral volume, etc.

 

[HallsofIvy]

If you are given that a region has height z= f(x,y) at each point in region R in the xy-plane, its volume is given by
[tex]\int_R\int f(x,y)dxdy[/itex]

If it happens that z= f(x,y)= 1 for all x and y, that volume is just the area of R times 1. In other words, the area of R is
$$\int_R\int dxdy$$.