Understanding the meaning of multiple integrals

[Dr. Philgood]

Homework Statement

I am currently taking calc III and we have starting getting into double and triple integrals. I was wondering what you are actually doing when you take a double or triple integral? And what the difference is. I understand that you find area with a single integral and find volume with a double, but you can also find find volume with a triple integral. While there are some things such as mass that can only be found using a triple integral.

I guess my quest is: what are you actually doing to an object when you take multiple integrals of it?

 

[tiny-tim]

Welcome to PF!

Hi Dr. Philgood! Welcome to PF!

what are you actually doing to an object when you take multiple integrals of it?

You're dividing it into tiny cubes (tiny in all three directions), pretending the integrand is constant in each cube, and adding the values for all the cubes.

Sometimes you divide it into thin rods (tiny in two directions), and then it's only a double integral.

And sometimes you divide it into thin slices (discs, cylindrical shells, etc) (tiny in one direction), and then it's only a single integral.

You can get away with that if the integrand is approximately constant over each rod or slice. ​

 

[Dr. Philgood]

Okay, so you can think of going from a single integral to double integral as instead of adding squares adding cubes from a limit. For instance a single integral should be 2D while a double integral is 3D. That made it a lot easier to understand. Thank you.

 

[genericusrnme]

The easiest way I've seen is to just remember that the integral is the limit as all the dx's go to zero and a sum over them goes to infinity

so your integral over f(x,y,z)dxdydz is a sum over all the little cubes of size dxdydz

Okay, so you can think of going from a single integral to double integral as instead of adding squares adding cubes from a limit. For instance a single integral should be 2D while a double integral is 3D. That made it a lot easier to understand. Thank you.

When you are doing a single integral, you are summing over little lengths dx, you're adding up the value of f[x] multiplied by the small length dx (this gives you the area of a rectangle over the length dx)
When you do a double integral you are summing over little flat squares of sides dxdy, this can be taken to be a volume, just as the single integral can be taken to be an area, if you make your f(x,y) be the 'height' then you are summing over little parallelepiped (I hope I spelled that correctly) of height f(x,y) and base dx dy.
When you take a triple integral you are summing over little cubes of sides dxdydz, this again can be interprited as a 'volume' integral for some higher dimension, you could take the mass to be the 'volume' under the graph of the density function.
But human brains aren't wired to let us think about anything other than 3 dimensions.

I hope you understand what I mean and I haven't just made it seem more complicated :p