axis

Examples of axis in the following topics:

• Cylindrical Shells

  • In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
  • The volume of the solid formed by rotating the area between the curves of $f(x)$ and $g(x)$ and the lines $x=a$ and $x=b$ about the $y$-axis is given by:
  • The volume of solid formed by rotating the area between the curves of $f(y)$ and and the lines $y=a$ and $y=b$ about the $x$-axis is given by:
  • Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.
  • Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.

• Area of a Surface of Revolution

  • A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis .
  • Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis.
  • 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 the $y$-axis, then the area $A_y$ is given by the integral:
  • Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:
  • A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).

• Volumes of Revolution

  • The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.
  • The shell method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.
  • Disc integration about the $y$-axis.
  • Integration is along the axis of revolution ($y$-axis in this case).
  • The integration (along the $x$-axis) is perpendicular to the axis of revolution ($y$-axis).

• Cylindrical and Spherical Coordinates

  • A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.
  • 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.
  • For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian $xy$-plane (with equation $z = 0$), and the cylindrical axis is the Cartesian $z$-axis.
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• Double Integrals Over General Regions

  • the projection of $D$ onto either the $x$-axis or the $y$-axis is bounded by the two values, $a$ and $b$.
  • any line perpendicular to this axis that passes between these two values intersects the domain in an interval whose endpoints are given by the graphs of two functions, $\alpha$ and $\beta$.
  • $x$-axis: If the domain $D$ is normal with respect to the $x$-axis, and $f:D \to R$ is a continuous function, then $\alpha(x)$  and $\beta(x)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
  • $y$-axis: If $D$ is normal with respect to the $y$-axis and $f:D \to R$ is a continuous function, then $\alpha(y)$ and $\beta(y)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
  • In this case the two functions are $\alpha (x) = x^2$ and $\beta (x) = 1$, while the interval is given by the intersections of the functions with $x=0$, so the interval is $[a,b] = [0,1]$ (normality has been chosen with respect to the $x$-axis for a better visual understanding).

• Polar Coordinates

  • Polar coordinates define the location of an object in a plane by using a distance and an angle from a reference point and axis.
  • A positive angle is usually measured counterclockwise from the polar axis, and a positive radius is in the same direction as the angle.
  • A negative radius would be opposite the direction of the angle and a negative angle would be measured clockwise from the polar axis.
  • The polar axis is usually drawn horizontal and pointing to the right .
  • Use a polar coordinate to define a point with $r$ (distance from pole), and $\theta$(angle between axis and ray)

• Curve Sketching

  • If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve.
  • Similarly, if the exponent of $y$ is always even in the equation of the curve, then the $x$-axis is an axis of symmetry for the curve.

• Vectors in Three Dimensions

  • For instance, in three dimensions, the points $A=(1,0,0)$ and $B=(0,1,0)$ in space determine the free vector $\vec{AB}$ pointing from the point $x=1$ on the $x$-axis to the point $y=1$ on the $y$-axis.
  • Thus the bound vector represented by $(1,0,0)$ is a vector of unit length pointing from the origin along the positive $x$-axis.

• The Definite Integral

  • A definite integral is the area of the region in the $xy$-plane bound by the graph of $f$, the $x$-axis, and the vertical lines $x=a$ and $x=b$.
  • Given a function $f$ of a real variable x and an interval $[a, b]$ of the real line, the definite integral $\int_{a}^{b}f(x)dx$ is defined informally to be the area of the region in the $xy$-plane bound by the graph of $f$, the $x$-axis, and the vertical lines $x = a$ and $x=b$, such that the area above the $x$-axis adds to the total, and that the area below the $x$-axis subtracts from the total.

• Planetary Motion According to Kepler and Newton

  • The square of the orbital period is proportional to the cube of the semi-major axis of the planet's orbit.
  • The important components of an ellipse are as follows: semi-major axis $a$, semi-minor axis $b$, semi-latus rectum $p$, the center of the ellipse, and its two foci marked by large dots.