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Feldoh
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I heard that Archimedes proved geometrically that the area under the curve of x^2 is equal to x^3/3. I was just wondering if anyone could give me a link to the proof or try and explain it.
Thanks^^
Thanks^^
neutrino said:If you can get hold of a copy, read the intro chapter to Apostol's Calculus Vol.1.
Archimedes' area approximation for x^2 is a geometric method used to calculate the area under a curve. By inscribing and circumscribing regular polygons within a circle, Archimedes was able to estimate the area of the circle and, by extension, the area under a parabola. The more polygons used, the more accurate the approximation becomes.
Archimedes' method is useful because it provides a way to approximate the area under a curve without using calculus. This can be helpful in situations where calculus is not applicable, or when a quick estimation is needed.
The formula for calculating the area under a parabola using Archimedes' method is A = (3/4) * (b^2), where A is the area and b is the length of the base of the inscribed and circumscribed polygons.
The accuracy of Archimedes' method depends on the number of polygons used. The more polygons used, the closer the approximation will be to the actual area. Archimedes himself was able to achieve an accuracy of three decimal places using this method.
Yes, Archimedes' method can be used to calculate the area under other curves as long as they can be inscribed and circumscribed by regular polygons. However, the formula for calculating the area will vary depending on the shape of the curve.