What is the purpose of the integrand in triple integration?

[xphysics]

Hi... So I've been self-teaching Calculus III and I'm currently having a hard time coping with the idea of triple integration. You know how the integrand is f(x,y,z)? isn't that the equation to represent a 4D sketch? because technically, f(x,y,z) is ANOTHER VARIABLE and therefore giving us a 4 variable equation? If you do a double integral, you would have f(x,y) and then that would be a 3 dimensional shape; therefore, I could imagine it. BUT HOW COULD I IMAGINE A 4D SHAPE?

ALSO, I'VE BEEN LEARNING THIS THROUGH NOTES ONLINE SO BEFORE YOU CONSIDER THE FIRST QUESTION, PLEASE ANSWER THIS STUPID QUESTION, WHAT DOES THE EQUATION IN THE INTEGRAND REPRESENT? I KNOW WHAT THE LIMITS REPRESENT BUT NOT THE INTEGRAND. I THOUGHT IT REPRESENTS THE SURFACE TO INTEGRATE WITH WHEN TIMES BY dxdy/dxdydz

source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx

I tried to look it up, they said f(x,y,z) represents the density at that point, and when it times by dydzdx, basically it's density times volume... you'll get mass?

What!? can someone explain this please?

 

[verty]

If total mass M is distributed over a volume, we can represent how that mass is distributed with a function f(x,y,z) that gives the density at point (x,y,z). Now integrate over the volume and you regain the total mass. This is what the triple integral represents, it maps from a region of the density field to the total mass in that region.

I think this is perfectly clear, probably your book said exactly the same thing. If it didn't, I apologize, but what you should now do is read it again and try to understand it, and I want to say that this forum is surely not meant to be a substitute for thinking on one's own.

 

[xphysics]

If total mass M is distributed over a volume, we can represent how that mass is distributed with a function f(x,y,z) that gives the density at point (x,y,z). Now integrate over the volume and you regain the total mass. This is what the triple integral represents, it maps from a region of the density field to the total mass in that region.

I think this is perfectly clear, probably your book said exactly the same thing. If it didn't, I apologize, but what you should now do is read it again and try to understand it, and I want to say that this forum is surely not meant to be a substitute for thinking on one's own.

Again, I'm not learning from the book, I was just self-teaching through online notes from the one i showed... and i already articulated that I grasped the idea of f(x,y,z) represent density, it's just that if it is.. then the integration usage is completely different compare to single or double integral right? i mean when f(x,y) doesn't represent density, it represents the surface area for integration; same goes for f(x) you just draw a line to find the area underneath the curve.

 

[haruspex]

the integration usage is completely different compare to single or double integral right? i mean when f(x,y) doesn't represent density, it represents the surface area for integration; same goes for f(x) you just draw a line to find the area underneath the curve.

There's no specific physical meaning implied by the concept of integration, in however many dimensions. Certain physical applications can help to visualise the process. For a single or double integral, it is convenient to think of an area or a volume. But you could also have, say, a 2-D plate with varying area density. Likewise, a triple integral could be to find the volume of a hypersolid, but it's easier to imagine it as finding the mass of a 3-D object with varying density.
Note also that finding a volume can be as a double or triple integral: ∫∫∫dxdydz or ∫∫z(x,y)dydz.

 

[verty]

Note also that finding a volume can be as a double or triple integral: ∫∫∫dxdydz or ∫∫z(x,y)dydz.

Or with a single integral by integrating the sectional area. It's all a matter of taste.