1, the formula for finding the area and volume of a circle. One of Archimedes' most important achievements in mathematics is to find a formula to calculate the area and volume of a circle. He found that the area of a circle equals πr? Where r is the radius of the circle. This formula is one of the foundations of modern mathematics and is widely used in geometry, algebra and physics.

2. Infinite series. Another important achievement of Archimedes in mathematics is infinite series. He used infinite series to calculate some mathematical constants, such as pi and the base e of natural logarithm. Archimedes method provides a new calculation method for later mathematicians, which can be used to calculate some accurate values that cannot be obtained by conventional methods.

3. calculus. Archimedes' third important achievement in mathematics was calculus. Although Archimedes was not the inventor of calculus, he made an important contribution to the development of calculus. He discovered many basic principles of calculus, such as the basic concepts and calculation methods of derivatives and integrals.

Application of calculus

1, the application of natural science. Calculus is widely used in natural science. For example, in physics, calculus is used to describe the laws of motion of objects. For example, Newton's second law, F=ma, involves the differentiation of acceleration. In chemistry, calculus can be used to describe the relationship between chemical reaction rate and substance concentration.

2. Application in social science. Calculus has been widely used in social sciences. For example, in economics, calculus is used to analyze the optimization of cost, benefit and utility. In social science, calculus can be used to study the quantitative analysis and prediction of social phenomena, such as population growth and market prediction.

3. Applications in the field of engineering and technology. The application of calculus in engineering technology is direct. For example, in mechanical engineering, calculus is used to study the mechanical properties and motion state of objects. In electronic engineering, calculus can be used to analyze the current and voltage changes of circuits. In civil engineering, calculus can be used to study the stability and vibration of structures.