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I have some conceptual problems regarding multiple integrals,out of whichsomeoften make me do sums wrong. Please help me out!

1. If we triple integrate a function f(x,y,z) within appropiate limits (there are some sums of this kind in my book) are we integrating in 4-dimensions?---(since a function of x,y and z can only be described in 4 dimensions)......if so,integrations of higher order must involve higher dimensions!!

2. I found a sum (involving cylindrical coordinates),in which z is integrated within the limits r and r^2...I suppose this is just a numerical point of view,as z does not take the values of r....am I right?

3. Suppose we integrate a function xy ( limits are x: 0 to y and y:0 to 1), it says in my book,that we can separately integrate x and y (within appropiate limits), and then multiply the separate answers...but I don't understand how this should give us the right answer.....afterall, each of the integrals will have separate (constant) answers,and multiplying them will give thearea of a rectangle(with those dimensions),and not the rqd area under the curves.

4. In my book, it says that when we're double integrating a function,we sum up all the insinitesimal units (dx or dy) in one direction,and then in the other direction.....now,we usually find the integral by integrating between variable limits first and then for the 2nd integration step,we take constant limits........if we did it in theopposite order,we get an answer with a variable in it...but there's nothing conceptually wrong in that,is there?

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# Multiple integrals

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