Questions about double and triple integrals

[Maurice7510]

Hey,
I was just going through my vector calc textbook for this year and everything was going well until I reached double and triple integrals. My problem is the whole symmetry thing; when does (forgive me, I can't figure out the symbols) the integral from a to b become twice the integral from 0 to a versus becoming zero?
Other than that, I think I was doing fine, but if you guys wouldn't mind posting some double and triple integral questions for me so I could get some practice that would be great.
Thanks,
Maurice

 

[HallsofIvy]

Much the same thing as you saw with single integrals: if f(x) is an even function, then \int_{-a}^a f(x)dx= 2\int_0^a f(x)dx and if f(x) is an odd function, then \int_{-a}^a f(x)dx= 0. More generally, if f(x,y,z) is exactly the same in two regions, then the integral over the two is just twice the integral over one:I+ I= 2I. If f(x,y,z) is the same in two regions, except that one is the negative of the other, they cancel and the integral is 0: I- I= 0.

 

[Maurice7510]

That helps but there's a couple things: even fcn is where -f(x) = f(x) and odd fcn is where -f(x) = f(-x)? I just kind of forget, sorry. Also, what always bothered me about this is that, for example, if you have a fcn where it's graph is symmetrical with one region negative but equal to the other, how does the area "cancel out"?

 

[Maurice7510]

I think another problem I'm having is interpreting the integrals. For example, there's a problem I'm looking at in the textbook, where it asks us to calculate the triple integral of y^2 of, and this is the hard part for me, the tetrahedron in the first octant bounded by the coordinate planes and the plane 2x+3y+z=6. I have no idea how to choose my limits of integration, i.e. integrate from what to what?

If somebody can explain how to choose these, that would be greatly appreciated.
Thanks again,
Maurice