Chenming Hu 1797365 / Qiwei Liang 33196163

Alphonse Wang 47168166 / Yuqi Liu 44616167

Motivation of Integration

Before 370 BC, ancient Creek Astronomer Eudoxus had an idea to find the specific areas and volumes by breaking them up into small pieces of area for which the area and volume was known. This method was further developed and employed by Archimedes in 3rd century BC and used to calculate approximation to the area of a circle. And over last thousand years, the way of using integral has been promoted by lots of scientist. During these thousand years’ promotion, a famous scientist Isaac Newton used a small vertical bar above a variable to indicate integration. However, the box notation was difficult for printers to reproduce, so these notations were not widely adopted. Finally, in 1675 Gottfried Leibniz create the modern notation for the indefinite integral.

Today ‘Integration’ as a term is widely applicated as different meaning in different disciplines such as Sociology, economy and Engineering. Etc. In mathematics, ‘integration’ usually is represented as ‘integral’ which utilize a function to assign number to represent area, volume and other concepts by breaking them up into an infinite number of division. For example, using integrals to find out the different value. For some functions, some parts are positive and above the x-axis, and some parts are negative which are below the x-axis. When we try to figure out the average of these values, we can use the graph, and add the positive areas and negative areas together and find out the difference between two values. Using graph can give a visual feeling directly and help researchers.

Moreover, people use ‘integral’ to determine the area under a curve and do the data analyzing in many field, typically, in physics. In physics, people always use a function to represent the relationship between speed and time. From this kind of function, we are not only looking the changing of the speed, but also the area under the curve, which can represent the displacement of the object in this time period. When we analyze the function, we can find out the moving state and position state of this object. What is more, when the function is about the acceleration and time. The area under the curve is about the speed, and by analyzing the acceleration and speed of the object, we can figure out the position. Therefore, the data can help us find more data. Both of these two examples relate the integral and the real life together, and it shows the integrals exist in many fields.

All in all, integration has already connected with our life and acadamic study. So hopefully, right now you have the motivation to learn ‘INTEGRATION’.