What does the triple integral of a function represent graphically in 4D?

[GRB 080319B]

If \int\int\int_{Q}dV = Volume_{Q}, and graphically, it represents the volume between all the boundaries of the respective variables in the iterated integral, what does \int\int\int_{Q}f(x,y,z)dV represent? Does this integral represent the (volume?) "above" (in 4D sense) the solid represented by the boundaries and "under" w = f(x,y,z)? I'm using this type of integral for centers of mass and moments of inertia and having trouble visualizing it. Thanks.

 

[HallsofIvy]

"Volume", "center of mass" and "moments of intertia" are all applications of integration. An integral by itself does not necessarily "represent" anything!

IF f(x,y,z) is the mass density at the point (x, y, z), THEN \int\int\int f(x,y,z)dxdydz is the mass of the object.

 

[GRB 080319B]

Thanks for the reply, and I'm sorry if I misspoke. In the book I'm using now (Multivariable Calculus, Larson 8th Ed.), it visualizes the process of integration by viewing the iterated integral of the triple integral in terms of three sweeping motions, each adding another dimension to the solid region. I'm wondering, once you have this solid region, like the region beneath z = f(x,y) in the double integrals, is there any point in trying to visualize what's going on between this solid region and w = f(x,y,z)? I think I'm having trouble conceptualizing how things are working in 4D. If I'm still thinking about this wrong, feel free to correct me. Thanks.