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!