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  2. Goat grazing problem - Wikipedia

    en.wikipedia.org/wiki/Goat_grazing_problem

    Find the area between a circle and its involute over an angle of 2 π to −2 π excluding any overlap. In Cartesian coordinates, the equation of the involute is transcendental; doing a line integral there is hardly feasible. A more felicitous approach is to use polar coordinates (z,θ).

  3. Polar coordinate system - Wikipedia

    en.wikipedia.org/wiki/Polar_coordinate_system

    Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point ( r , φ ) can be expressed with an infinite number of different polar coordinates ( r , φ + n × 360°) and (− r , φ + 180° + n × 360°) = (− r , φ ...

  4. Area of a triangle - Wikipedia

    en.wikipedia.org/wiki/Area_of_a_triangle

    When working in polar coordinates it is not necessary to convert to Cartesian coordinates to use line integration, since the line integral between consecutive vertices (r i,θ i) and (r i+1,θ i+1) of a polygon is given directly by r i r i+1 sin(θ i+1 − θ i)/2. This is valid for all values of θ, with some decrease in numerical accuracy ...

  5. Del in cylindrical and spherical coordinates - Wikipedia

    en.wikipedia.org/wiki/Del_in_cylindrical_and...

    This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.

  6. Planimeter - Wikipedia

    en.wikipedia.org/wiki/Planimeter

    The connection with Green's theorem can be understood in terms of integration in polar coordinates: in polar coordinates, area is computed by the integral (()), where the form being integrated is quadratic in r, meaning that the rate at which area changes with respect to change in angle varies quadratically with the radius.

  7. Area of a circle - Wikipedia

    en.wikipedia.org/wiki/Area_of_a_circle

    Since the area of the rectangle is ab, the area of the ellipse is π ab/4. We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk.

  8. Coordinate system - Wikipedia

    en.wikipedia.org/wiki/Coordinate_system

    Another common coordinate system for the plane is the polar coordinate system. [7] A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line).

  9. Spherical coordinate system - Wikipedia

    en.wikipedia.org/wiki/Spherical_coordinate_system

    Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).