Математический анализ параметрических кривых: площади поверхностей

A parametric curve defines a path in the plane where x and y depend on a single parameter, usually time, ranging from alpha to beta.

When the curve rotates around an axis, it sweeps out a three-dimensional surface.

This is called a surface of revolution. Its surface area depends on the curve’s shape and its distance from the axis of rotation.

Consider a circle of radius r, centered at a distance R from the y-axis. Its x and y coordinates are given parametrically for t from zero to two pi. Rotating this circle around a non-intersecting axis in the same plane creates a torus.

To calculate the torus surface area in parametric form, consider an infinitesimal strip on the surface. As a point on the circle rotates, it traces a circular path. Its circumference gives the strip’s length. The strip’s width comes from the arc-length differential of the parametric curve.

Multiplying the two gives the differential area, and integrating this over the parameter t gives the total surface area in parametric form.

This method helps design components like O-rings, where an exact surface area helps set the size and contact area to stop leaks in machines.