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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 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.
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.
The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height (the quadrature of the parabola); The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes;
The main objective of book two of On the Equilibrium of Planes is the determination of the centre of gravity of any part of a parabolic segment, as shown in Proposition 8. The book begins with a simpler proof of the law of the lever in Proposition 1, making reference to results found in Quadrature of the Parabola .
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 area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB. Proof: [1]: 15–18 Let D be the midpoint of AC. Construct a line segment JB through D, where the distance from J to D is equal to the distance from B to D. We will think of the segment JB as a "lever" with D as its fulcrum. [3]
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.