What is the Integral and Differential calculus useful for?

The Integral and Differential calculus is one of the most important branches of Mathematics. Its aim is to study the variation rate of quantities such as the Mean Rate of variation of a function which stands for the sales of an enterprise throughout the times. Besides that, it is also used to calculate the Accumulation of Quantities such as the previous referred area below the function of sales, which stands for the total of sales accumulated by the enterprise throughout years.

Who has developed the Integral and Differential calculus?

Its development results from the study and research made by several mathematicians throughout the centuries. But the greatest contribution to its spread comes from the work of two great mathematicians who, in spite of working separately, arrived to the same conclusions almost at the same time: Isaac Newton and Gottfried Leibniz.

What is the Riemann Sum?

At the beginning, when the concept of Integral emerged, it aimed at calculating the area below a slope in a Cartesian Referential. The German Mathematician Bernhard Riemann gave rise to a method which allows the calculation of an area of the plane region which is placed between a certain gap (between two values). The basic idea consists of dividing the area into smaller and smaller rectangles and then calculating the area of each one of them. The smaller the rectangles are the closer to the real value we will be.

Could you give a practical example of its usefulness?

If you have a look at the picture below, you will be able to notice the wonderful architectural work of the "Pavilhão Atlântico" roof that can be found in Lisbon. This roof has a certain thickness and we are sure that it was necessary to calculate the area under that slope in order to get its mass. It wouldn’t be possible without the wonderful help of the Integral Calculus! Even if today it is argued that nobody does it by hand anymore. It’s enough to introduce the data in the computer and it will return us the values. Thus, it was necessary someone who knows about Integral Calculus to programme the computer so that it can be able to reach the right value.

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