Arc length and surface of revolution

A curve in, say, the plane is approximated by connecting the finite number of points on curve using line segments to create the polygonal path. As it is straightforward to compute the length of each linear segment the total length of approximation is found by summing up the lengths of each linear segment.

If the curve is not a polygonal path, better approximations to the curve can be obtained by following the shape of the curve increasingly much  closely. The approach is increasingly used as larger number of segments of smaller lengths. The lengths of the successive approximations do not decrease and will eventually keep increasing-possibly indefinitely, but for the smooth curves this will tend to a limit as the lengths of the segments get arbitrarily small.

For some of the curves there is a smallest number L that is an upper bound on the length of any polygonal approximation. If such a number exists, then curve is said to be rectifiable and the curve is defined to have arc length L. A curve may be parameterized in a number of ways. Therefore the arc length is an intrinsic property of the curve, meaning that it does not depend on the choice of parameterization. The definition of arc length for the curve is analogous to definition of total variation of a real-valued function. A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane (the axis)

Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is coplanar with the axis, as well as the hyperboloids of 1 sheet when line is skew to the axis. A circle which is rotated about the diameter generates a sphere and if the circle is rotated about the coplanar axis other than the diameter it generates a torus. A surface of revolution is a surface created by rotating a curve lying on some plane around a straight line which lies on the same plane. The total length of the whole arc can then be obtained by adding up all lengths of small arc pieces in Riemann sum sense under the limiting process.

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