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Archimedes used a method known as the "method of exhaustion" to find the area under a parabola. This involved inscribing a series of polygons within the parabola and calculating their areas. By increasing the number of sides of the polygons, Archimedes was able to get closer and closer to the exact area under the parabola.
Archimedes' discovery of the quadrature of the parabola without using calculus was significant because it demonstrated the power and versatility of geometry and the "method of exhaustion" in solving complex mathematical problems. It also paved the way for future advancements in calculus and other branches of mathematics.
Archimedes' method of exhaustion is considered a precursor to modern calculus. While it does not use the same principles and techniques as calculus, it achieves a similar result by approximating the area under a curve. It also laid the foundation for later developments in integral calculus.
In addition to the quadrature of the parabola, Archimedes used the "method of exhaustion" to solve problems such as finding the area of a circle, the volume of a sphere, and the surface area of a cylinder. He also used this method to determine an accurate value for pi.
Archimedes' contributions to mathematics and science were significant and far-reaching. His use of the "method of exhaustion" influenced the development of calculus and other mathematical techniques. His work also had an impact on fields such as physics and engineering, and his discoveries continue to be studied and applied in modern research and technology.
