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History of Calculus

History of Calculus. CURVES AREAS VOLUMES. METHOD OF EXHAUSTION. Discovered by Antiphon Inscribe a shape with multiple polygons whose area converge to the area of the shape. Eudoxus. Eudoxus rigorously developed Antiphon’s Method of Exhaustion

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History of Calculus

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  1. History of Calculus CURVES AREAS VOLUMES

  2. METHOD OF EXHAUSTION • Discovered by Antiphon • Inscribe a shape with multiple polygons whose area converge to the area of the shape

  3. Eudoxus • Eudoxus rigorously developed Antiphon’s Method of Exhaustion •  For the calculation of the volume of the pyramid and cone. Archimedes notes that Eudoxus was the first to prove that the cone and the pyramid are one-third respectively of the cylinder and prism with the same base and height.

  4. Archimedes • Archimedes used the method of exhaustion to compute the area inside a circle

  5. Archimedes cont... Other results obtained by Archimedes using the method of exhaustion • 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 area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; • The volume of a sphere is 4 times that of a cone having a base and height of the same radius; • The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter; • The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length; • Use of the method of exhaustion also led to the successful evaluation of a geometric series (for the first time). Estimations of Pi were being worked out in ancient Greece

  6. Liu Hui(3rd Century AD) • Liu provided a detailed step-by-step description of an iterative algorithm to calculate pi to any required accuracy based on bisecting polygons; he calculated pi to between 3.141024 and 3.142708 with a 96-gon; he suggested that 3.14 was a good enough approximation, and expressed pi as 157/50; he admitted that this number was a bit small. Later he invented an ingenious quick method to improve on it, and obtained π ≈ 3.1416 with only a 96-gon, with an accuracy comparable to that from a 1536-gon.

  7. Gregoire de Saint-Vincent • Method of Exhaustion was expanded by Gregoire de Saint-Vincent when he discovered that the area under a rectangular hyperbola (i.e. a curve given by xy = k) is the same over [a,b] as over [c,d] when a/b = c/d. • Although a circle-squarer he is known for the numerous theorems which he discovered in his search for the impossible; Jean-ÉtienneMontucla ingeniously remarks that "no one ever squared the circle with so much ability or (except for his principal object) with so much success."

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