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A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. [1] [2] Nevertheless, for some figures a quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity.
A parabolic segment. SVG redraw of en:Image:Parabolic Segment.png: Date: 30 June 2008: ... Cavalieri's quadrature formula; Quadrature of the Parabola; Global file usage.
The area A of the parabolic segment enclosed by the parabola and the chord is therefore =. This formula can be compared with the area of a triangle: 1 / 2 bh. In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord.
The area of a segment of the parabola cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus. In medieval Europe the quadrature meant calculation of area by any method.
The term "quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve y = x n. Traditionally important cases are y = x 2, the quadrature of the parabola, known in antiquity, and y = 1/x, the quadrature of the hyperbola, whose value is a logarithm.
A FBI document obtained by Wikileaks details the symbols and logos used by pedophiles to identify sexual preferences. According to the document members of pedophilic organizations use of ...
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola. In Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4 / 3 times the area of a corresponding inscribed triangle as ...