cylinder

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: