axis

(noun)

a fixed, one-dimensional figure, such as a line or arc, with an origin and orientation and such that its points are in one-to-one correspondence with a set of numbers; an axis forms part of the basis of a space or is used to position and locate data in a graph (a coordinate axis)

Related Terms

  • curve
  • area

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)f(x)f(x) and g(x)g(x)g(x) and the lines x=ax=ax=a and x=bx=bx=b about the yyy-axis is given by:
    • The volume of solid formed by rotating the area between the curves of f(y)f(y)f(y) and and the lines y=ay=ay=a and y=by=by=b about the xxx-axis is given by:
    • Each segment located at xxx, between f(x)f(x)f(x)and the xxx-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)x(t)x(t), y(t)y(t)y(t), with ttt ranging over some interval [a,b][a,b][a,b] and the axis of revolution the yyy-axis, then the area AyA_yA​y​​ is given by the integral:
    • Likewise, when the axis of rotation is the xxx-axis, and provided that y(t)y(t)y(t) is never negative, the area is given by:
    • A portion of the curve x=2+coszx=2+\cos zx=2+cosz rotated around the zzz-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 yyy-axis.
    • Integration is along the axis of revolution (yyy-axis in this case).
    • The integration (along the xxx-axis) is perpendicular to the axis of revolution (yyy-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 xyxyxy-plane (with equation z=0z = 0z=0), and the cylindrical axis is the Cartesian zzz-axis.
    • A cylindrical coordinate system with origin OOO, polar axis AAA, and longitudinal axis LLL.
  • Double Integrals Over General Regions

    • the projection of DDD onto either the xxx-axis or the yyy-axis is bounded by the two values, aaa and bbb.
    • 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, α\alphaα and β\betaβ.
    • xxx-axis: If the domain DDD is normal with respect to the xxx-axis, and f:D→Rf:D \to Rf:D→R is a continuous function, then α(x)\alpha(x)α(x)  and β(x)\beta(x)β(x) (defined on the interval [a,b][a, b][a,b]) are the two functions that determine DDD.
    • yyy-axis: If DDD is normal with respect to the yyy-axis and f:D→Rf:D \to Rf:D→R is a continuous function, then α(y)\alpha(y)α(y) and β(y)\beta(y)β(y) (defined on the interval [a,b][a, b][a,b]) are the two functions that determine DDD.
    • In this case the two functions are α(x)=x2\alpha (x) = x^2α(x)=x​2​​ and β(x)=1\beta (x) = 1β(x)=1, while the interval is given by the intersections of the functions with x=0x=0x=0, so the interval is [a,b]=[0,1][a,b] = [0,1][a,b]=[0,1] (normality has been chosen with respect to the xxx-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 rrr (distance from pole), and θ\thetaθ(angle between axis and ray)
  • Curve Sketching

    • If the exponent of xxx is always even in the equation of the curve, then the yyy-axis is an axis of symmetry for the curve.
    • Similarly, if the exponent of yyy is always even in the equation of the curve, then the xxx-axis is an axis of symmetry for the curve.
  • Vectors in Three Dimensions

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

    • A definite integral is the area of the region in the xyxyxy-plane bound by the graph of fff, the xxx-axis, and the vertical lines x=ax=ax=a and x=bx=bx=b.
    • Given a function fff of a real variable x and an interval [a,b][a, b][a,b] of the real line, the definite integral ∫abf(x)dx\int_{a}^{b}f(x)dx∫​a​b​​f(x)dx is defined informally to be the area of the region in the xyxyxy-plane bound by the graph of fff, the xxx-axis, and the vertical lines x=ax = ax=a and x=bx=bx=b, such that the area above the xxx-axis adds to the total, and that the area below the xxx-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 aaa, semi-minor axis bbb, semi-latus rectum ppp, the center of the ellipse, and its two foci marked by large dots.
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