Examples of axis in the following topics:
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- 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.
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- 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 Ay 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+cosz rotated around the z-axis (vertical in the figure).
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- 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).
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- 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.
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- 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, α and β.
- x-axis: If the domain D is normal with respect to the x-axis, and f:D→R is a continuous function, then α(x) and β(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→R is a continuous function, then α(y) and β(y) (defined on the interval [a,b]) are the two functions that determine D.
- In this case the two functions are α(x)=x2 and β(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).
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- 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 θ(angle between axis and ray)
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- 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.
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- For instance, in three dimensions, the points A=(1,0,0) and B=(0,1,0) in space determine the free vector AB⃗ 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.
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- 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 ∫abf(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.
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- 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.