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If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y-axis, then the surface area A y is given by the integral = () + (), provided that x(t) is never negative between the endpoints a and b.
A parametric surface is a surface in the Euclidean space ... This is true for a circular cylinder, sphere, cone, torus, and a few other surfaces of revolution.
Rotating a curve. The surface formed is a surface of revolution; it encloses a solid of revolution. Solids of revolution (Matemateca Ime-Usp)In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).
A surface of revolution is obtained by rotating a curve in the xz-plane about the z-axis. Such surfaces include spheres, cylinders, cones, tori, and the catenoid. The general ellipsoids, hyperboloids, and paraboloids are not. Suppose that the curve is parametrized by = (), = ()
A parametric surface need not be a topological surface. A surface of revolution can be viewed as a special kind of parametric surface. If f is a smooth function from R 3 to R whose gradient is nowhere zero, then the locus of zeros of f does define a surface, known as an implicit surface. If the condition of non-vanishing gradient is dropped ...
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings , or more generally, of an affine transformation .
Philadelphia Eagles' A.J. Brown speaks during an NFL football news conference in Philadelphia, Thursday, Jan. 30, 2025, ahead of Super Bowl LIX against the Kansas City Chiefs.
The surface area of an ellipsoid of revolution ... The key to a parametric representation of an ellipsoid in general position is the alternative definition: