geometric interpretation of double integrals

[kelly070280]

If f(x,y) is greater than or equal to 0 on a region R in the plane, then the double integral on R of f(x,y)dA can be interpreted geometrically as the volume of the solid under the surface z=f(x,y) and above R.


If f(x,y)=1 for all (x,y) on a region R, where R has a nice shape, such as a rectangle, triangle, etc., what is another geometric interpretation of the double integral on R of f(x,y)dA?

Would it be something about finding the area?

Thanks!

 

[JakeD]

If f(x,y) is greater than or equal to 0 on a region R in the plane, then the double integral on R of f(x,y)dA can be interpreted geometrically as the volume of the solid under the surface z=f(x,y) and above R.


If f(x,y)=1 for all (x,y) on a region R, where R has a nice shape, such as a rectangle, triangle, etc., what is another geometric interpretation of the double integral on R of f(x,y)dA?

Would it be something about finding the area?

Yes.

Take the shape to be a circle, for example. The double integral is the volume of a cylinder, which is the height times the area of the circle. When the height is 1, the volume is just equals the area.

 

[kelly070280]

Thank you!