Double Integrals - Volume vs. Area

[Gramma2005]

I am confused about when a double integral will give you an area, and when it will give you a volume. Since we are integrating with respect to two variables, wouldn't that always give us an area? Don't we need a third variable in order to find the volume? Thanks for the help.

 

[nazzard]

Hello Gramma2005,

it depends on the function you are integrating.

Let's take a look at the function $$f(r)=4\pi r^2$$.

Yet the following integral, (only integrating with respect to one variable!) can be interpreted as the function for the volume of a sphere depending on the radius r.

$$F(r)=\int_{0}^{r} f(r') dr'$$

Regards,

nazzard

 

[HallsofIvy]

A double integral will give you an area when you are using it to do that!
A double integral is simply a calculation- you can apply calculations to many different things.

I think that you are thinking of the specific cases
1) Where you are given the equations of the curves bounding a region and integrate simply dA over that region. That gives the area of the region.

2) Where you are also given some height z= f(x,y) of a surface above a region and integrate f(x,y)dA over that region. That gives the volume between the xy-plane and the surface f(x,y). It should be easy to determine whether you are integrating dA or f(x,y)dA!

But that is only if f(x,y) really is a height. My point is that f(x,y) is simply a way of calculating things and what "things" you are calculating depends on the application. Sometimes a double integral gives pressure, sometimes mass, etc., depending on what the application is.