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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.
Hence, the area of the parabola must be 1/3 to give it the opposite torque. This type of method can be used to find the area of an arbitrary section of a parabola, and similar arguments can be used to find the integral of any power of x {\displaystyle x} , although higher powers become complicated without algebra.
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 :
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 ...
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 ...
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Archimedes proves the next seven propositions by combining the concept of centre of gravity and the properties of the parabola with the results previously found in On the Equilibrium of Planes I. Specifically, he infers that two parabolas that are equal in area have their centre of gravity equidistant from some point, and later substitutes ...