# How are integrals applied to everyday life

## Areas of application of integral calculus

Many who do integral calculus sometimes ask themselves: What for? But if integral and differential calculus were not important sub-areas of mathematics, they would not be dealt with, would they? There are some application exercises in math books, but mostly they just integrate and derive.

On the following pages we try to show clearly in which areas integral calculus is used.

Calculate the area between two curves. A classic that is taken through in every high school. But what is so interesting about this area?

First of all, it has to be said that curves can take many forms. So one could say that the world - that is, the objects that can be found around us - could be described in their form by mathematics. In most cases, however, these would no longer be simple functions, but rather highly complex and long. An example of such a complicated function comes straight from the world of comics: the **Bat curve.**

Now the question of the area of this extraordinary curve seems to be relatively uninteresting even for avowed Batman fans. But the Bat curve proves that there are no limits to complexity. Engineers have to calculate the areas of shapes for their constructions in the same way as manufacturers of products have to know how much of which materials is needed. Integral calculus can do this.

The calculation of volumes is at least as important as areas. Since the world around us is not flat like a flounder, but 3-dimensional, it often happens in real life that we have to calculate the volume of bodies. However, these are not ordinary bodies, they are created by rotating a surface through 360 °. That's why they will **Body of revolution** called. Rotational solids in mathematics are created similarly to figures on a lathe. An amazing number of objects can be made this way:

In addition to bowls, bowls and pepper mills, there are also other objects of rotation. The transmission shaft in a car can, for example, be mathematically described as a body of revolution. The calculation of the volume is of great importance from an engineering and economic point of view, because weight, stability and also the price depend on the nature and ultimately also the volume of the objects.

Of course, a lot is calculated in the natural sciences, especially in physics. It is therefore not surprising that the integral calculus is an indispensable companion there. In fact, there are so many areas of application for integral calculus in physics alone that only a few (very) few examples can be given here.

So it is not surprising that the invention of integral calculus is attributed to Gottfried Wilhelm Leibniz and Sir Isaac Newton - both were physicists. But what is so exciting about the area under a curve for physicists? The question is easy to answer for all those who have attended a physics course: If you have a function that describes the distance covered by an object, then the area under the curve is the speed of the object. The integral of the acceleration function, in turn, is the function for the speed. Other physical quantities have a similar relationship. Everything creates an elegant overall picture.

But integral calculus is not only the order of the day for physicists and engineers. All sciences that have mathematics as their descriptive language find areas of application in integral calculus. Even the economy. Because economics also know many models to mathematically describe the complex economic theories and models.

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