Last Updated on October 4, 2023 by Editorial Team

Have you ever wondered how the universe is constantly in motion and how it is monitored? Or how the motion of all the minute particles can be measured? The answer to all these curiosity questions lies in an interesting subject called Calculus. Calculus is the branch of math that studies the rate of change.

Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany independently developed calculus in the 17th century. It encompasses two concepts. The calculation of instantaneous rates of change is differentiation, and the summation of infinitely many small factors to determine a whole is termed integration.

Calculus helps to calculate and solve problems ranging from tracking the position of a satellite or predicting the pressure building up within a closed space. With important roles to play in such a range of applications, calculus is a very interesting concept.

**In the following post, let us understand what calculus is all about. **

**Integration and differentiation: Understanding the difference**

Calculus might seem very daunting for beginners. Jargon related to these crucial concepts can create ambiguity among learners. So, let us first understand the basic difference between the two as well as where these actually come into use in real life.

INTEGRATION | DIFFERENTIATION |

1. Integration sums up all small areas lying under a curve and determines the total area. | 1. Differentiation is the process by which the rate of change of a curve is determined. |

2. Integral calculus adds all the pieces together. | 2. Differential calculus deals with the process of dividing something to understand or calculate the changes. |

3. Integration deals with the distance traveled by the function and the area between the function and x-axis | 3. Differentiation calculates the speed and the slope of the function |

4. Integration is used to find out areas, volumes, and central points. | 4. Differentiation is used to determine if a function is increasing or decreasing, and the calculation of instantaneous velocity. |

**Application of integration and differentiation in real life: Practical examples**

Now that we know what differentiation and integration are all about, let us have a look at the roles they play in our real life.

**1. Calculating pressure within dams**

Let us consider a dam. When the reservoir behind it is full, the dam withstands a great deal of force. We can use integration to calculate the force exerted on the dam when the reservoir is full and also calculate how changing water levels affect that force. Hydrostatic force is one of the many applications of integrals.

**2. Automobiles**

In an automobile, we always find an odometer and a speedometer. Their gauges work in synchrony and determine the speed and distance the automobile has traveled. The electronic meters use differentiation to transform the data sent to the motherboard from the wheels (speed) and the distance (odometer).

**3. Business**

One of the most common applications of derivatives of differentiation is when data is computed on a graph or a data table like an Excel sheet. Once there is an input, one can calculate the profit or loss by means of derivatives.

**4. Astronomy**

Space flight engineers frequently use calculus when planning lengthy missions. To launch a rocket, calculus allows each of those variables to accurately take into account the orbiting velocities under the gravitational influences of the sun and the moon. This is one of the most crucial instances of real-life applications of differentiation.

**5. Economics**

Differential calculus is used in economics to calculate marginal cost, marginal revenue, maxima, and minima, and also enables economists to predict maximum profit (or) minimum loss in specific conditions.

**6. Swimming pools**

Integration determines the amount of water used to fill a swimming pool. We first need to determine the shape of the swimming pool and find its size. Therefore, we find the amount of water that will fill it. If the swimming pool shape is not a regular geometric shape, it begins with a slight gradient.

After which the slope descends steeply. The sides of the pool are either curved or semi-elliptical. The surface also has rises and falls. In such a case, it is possible to find the volume by using integration, which helps find the area between curved surfaces.

**7. Video games **

The graphic engineer uses integration and differentiation to determine the differences and changes in three-dimensional models and how they will change when exposed to multiple conditions. This helps to create a very realistic environment in 3D movies or video games.

**8. Medicine**

Calculus is a crucial mathematical tool for analyzing drug activity quantitatively. Differential equations are utilized to relate the concentrations of drugs in various body organs over time. In addition, integrated equations are often used to model the cumulative therapeutic or toxic outcomes of drugs in the body.

**9. Architecture**

Calculus can be used by architects to express design plans through graphs or drawings. They can describe surfaces through maths to help adapt the drawing to the computer software. This can be done through various differential equations. Calculus was used in the design and construction of the Eiffel Tower^{[1]}.

**10. Credit card**

Companies use differential calculus to calculate the minimum payable amount. There are several variables that go into this calculation. It is calculated by the amount of money that is due by the due date. The rate of interest also needs to be considered. With all these changing values, interest rates, and account balances, the calculation has to be done simultaneously in order to provide the customer with an accurate minimum balance and amount payable.

**Conclusion**

To sum up, calculus is a mathematical concept that is commonly utilized in mathematical models to get optimal solutions and helps in understanding the changes in the values associated with a function. Furthermore, the concepts of differentiation and integration play significant roles in our lives without us noticing them.

They are interrelated with math, science, economics, and almost every field of life. Differentiation and integration are not merely a part of a mathematical discipline but play a huge role in the real world. With such varied applications, calculus is the stepping stone for people aiming for a career in math or science.

**References**

[1] Weidman, P., & Pinelis, I. (2004). Model equations for the Eiffel Tower profile: historical perspective and new results. *Comptes Rendus Mécanique*, *332*(7), 571–584. https://doi.org/10.1016/j.crme.2004.02.021