cylinder

(noun)

a surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve

Related Terms

  • integration
  • volume
  • revolution

Examples of cylinder in the following topics:

  • Cylinders and Quadric Surfaces

    • A cylinder (from Greek "roller" or "tumbler") is one of the most basic curvilinear geometric shapes.
    • The surface is formed by the points at a fixed distance from a given line segment, the axis of the cylinder.
    • The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.
    • The surface area and the volume of a cylinder have been known since antiquity.
    • In common use, a cylinder is taken to mean a finite section of a right circular cylinder, i.e. the cylinder with the generating lines perpendicular to the bases, with its ends closed to form two circular surfaces.
  • Cylindrical Shells

    • The idea is that a "representative rectangle" (used in the most basic forms of integration, such as $\int x \,dx$) can be rotated about the axis of revolution, thus generating a hollow cylinder with infinitesimal volume.
    • Integration, as an accumulative process, can then calculate the integrated volume of a "family" of shells (a shell being the outer edge of a hollow cylinder), giving us the total volume.
    • By adding the volumes of all these infinitely thin cylinders, we can calculate the volume of the solid formed by the revolution.
  • Parametric Surfaces and Surface Integrals

    • The straight circular cylinder of radius $R$ about the $x$-axis has the following parametric representation: $\vec{r}(x,\phi)=(x,R \cos \phi, r \sin \phi)$.
  • Volumes of Revolution

    • Alternatively, where each disc has a radius of $f(x)$, the discs approach perfect cylinders as their height $dx$ approaches zero.
    • The lateral surface area of a cylinder is $2 \pi r h$, where $r$ is the radius (in this case $x$), and $h$ is the height (in this case $[f(x)-g(x)]$).
  • Cylindrical and Spherical Coordinates

    • Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on.
  • Triple Integrals in Cylindrical Coordinates

    • The function $f(x,y,z) = x^2 + y^2 + z$ is and as integration domain this cylinder:
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