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Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord. [3] Archimedes gives two proofs of the main theorem: one using abstract mechanics and the other one by pure geometry. In the first proof, Archimedes considers a lever in equilibrium under the action of gravity ...
Archimedes used the method of exhaustion to compute the area inside a circle. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a sequence of polygons with an increasing number of sides and a corresponding increase in area.
The area of the surface of a sphere is equal to quadruple the area of a great circle of this sphere. 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 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 shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1 / 4 :
This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a ...
The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere. The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment. For the proofs of these results, Archimedes used the method of exhaustion attributed to ...
Archimedes used the method of exhaustion to calculate the area under a parabola in his work Quadrature of the Parabola. Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus ( c. 390–337 BC ) developed the method of exhaustion to prove the formulas for cone and ...
Archimedes' theorem states that the total area under the parabola is 4/3 of the area of the blue triangle. His method was to dissect the area into infinite triangles as shown in the adjacent figure. [19] He determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and ...