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Topic: Volume of a curved body of rotation.

Friends! How to consider volume and the area of a body of rotation round a straight line, I know! But here it would be desirable to have the simple theory of volume and the area of a body of rotation with  an axis of central symmetry. I start to recall something about the curvilinear coordinate systems, divergence and tensors, but I feel that it should is somehow easier to dare for "rotation bodies" (or how here more correctly to formulate)! More shortly, there is a certain monotonous function f (x, y, z) in three-dimensional Euclidean space which serves as a local central axis of symmetry of a body of rotation round it. I.e. for each point with coordinates (x, y, z), any "normal" to a point f (x, y, z) intersects a body surface on equidistant distance ("radius") from this point. We admit radius such "rotation bodies" it is described by function r (x, y, z). I.e. we have two functions: f (.) - "An axis of central symmetry", and r (.) - Radius of a body of rotation in the given point f (.) . In the curvilinear coordinate systems I can count it somehow, but too it turns out ""! Whether There is no more simple method of calculation of the area and volume of a body for such narrow task? Something to me in a head, except "divergents" and "tensors" comes nothing... Or it does not turn out to simplify? Prompt, if there are ideas? Or with a muzzle into the link stick! Something I already absolutely "brake out of the blue"... Of Z.Y.integral I am not afraid!

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Re: Volume of a curved body of rotation.

Hello, maxluzin, you wrote: M> is shorter, there is a certain monotonous function f (x, y, z) in three-dimensional Euclidean space the Monotonous function of several variables!? Well it is necessary to write that you imply it - are not present after all in the mathematician of determination such (there are only different and nonequivalent determinations for any private problem domains). And further something too the strange is written smile Correctly I understand that the problem can be reformulated approximately so: in 3D it is parametric the curve c (t) is set. On this curve the center of a disk of variable radius r (t), thus the vector of the normal which have been let out from center moves, in each point concerns a curve. It is required to find volume swept by a disk. It is necessary? Then the directional volume will be such:

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Re: Volume of a curved body of rotation.

Hello, watchmaker, you wrote: W> Hello, maxluzin, you wrote: M>> is shorter, there is a certain monotonous function f (x, y, z) in three-dimensional Euclidean space W> the Monotonous function of several variables!? Well it is necessary to write that you imply it - are not present after all in the mathematician of determination such (there are only different and nonequivalent determinations for any private problem domains). And further something too the strange is written I do not know how in another way to formulate... Yes, and it would be desirable to tell that it is monotonous on all definition range, i.e. either strictly increases, or strictly decreases, without ruptures and is differentiated everywhere concerning a measure of Lebesgue. W> correctly I understand that the problem can be reformulated approximately so: in 3D it is parametric the curve c (t) is set. On this curve the center of a disk of variable radius r (t), thus the vector of the normal which have been let out from center moves, in each point concerns a curve. It is required to find volume swept by a disk. It is necessary? W> Then the directional volume will be such: W> Image: gif.latex Yes, it also is necessary! Thanks!