Real Life Applications of Calculus

Calculus is a powerful tool that can help us understand and solve many real-world problems. Despite its reputation as a difficult and abstract branch of mathematics, it has numerous applications in various fields such as engineering, physics, design, medicine, computer science, and data analytics. In this blog post, we will explore some of the amazing ways calculus is used to make our lives better.

What is Calculus?

The two main types of calculus, differential, and integral calculus, each have interpretations and applications in real life

• Differential calculus deals with the rate of change of a function. For example, if we want to know how fast a car is moving at a certain moment, we can use differential calculus to calculate speed to be the rate of change of position.
• Integral calculus deals with the total change or accumulation of a function over an interval. For example, if we want to know how far a car has traveled in a certain period of time, we can use integral calculus to find the distance traveled to be the integral of speed.

If you want to learn more about calculus, you can check out our blog post on What is Calculus

Physics

Physics is another field that heavily relies on calculus. Calculus helps us describe and predict the behavior of physical phenomena such as motion, gravity, light, sound, heat, electricity, and magnetism. Many principles of physics,  Some examples are:

• Kinematics: Calculus is used to study the motion of objects under various forces and conditions. For example, we can use calculus to find the position, velocity, and acceleration of a projectile launched at an angle with respect to time. We can also use calculus to find the optimal angle and speed for maximum range or height. For instance, if we want to throw a ball as far as possible on a flat ground with no air resistance, we can use calculus to find that the optimal angle is 45 degrees and the optimal speed is proportional to the square root of the acceleration due to gravity.
• Dynamics: Calculus is used to study the causes and effects of forces on objects. For example, we can use calculus to find the work done by a force on an object or the power generated by an engine. We can also use calculus to find the center of mass and moment of inertia of complex shapes. For example, if we want to find the center of mass of a triangular plate with uniform density and base b and height h, we can use calculus to find that it is located at (b/3,h/3) from one vertex.
• Thermodynamics: Calculus is used to study the relationship between heat, temperature, pressure and volume in various systems. For example, we can use calculus to find the entropy change or efficiency of a heat engine or a refrigerator. One example of a heat engine that uses calculus is the Carnot engine, which is the most efficient engine possible. The Carnot engine operates between two temperatures T1 and T2, and its efficiency is given by 1-T2/T1.
• Weather models: The National Weather Service uses a calculus-based computer program called the Global Forecast System (GFS) to produce weather forecasts for the entire world. The GFS solves a set of partial differential equations that describe the dynamics and thermodynamics of the atmosphere. The GFS uses numerical methods such as finite difference and finite element to approximate the solutions of these equations using calculus.

Engineering

One of the most important applications of calculus in real life is in engineering. Engineers use calculus to design and optimize structures, machines, systems, and processes. Some examples are:

• Structural engineering: Calculus is used to calculate heat loss in buildings, forces in complex structural configurations, and structural analysis in seismic design requirements. Architects use calculus to determine the amount of materials required for constructing support systems that can withstand stress over long periods of time. Even notable monuments such as the Eiffel Tower were constructed using calculus to predict the impact of wind resistance. The Eiffel Tower weighs about 10,000 tons, but its base only exerts a pressure of 4 kg/cm2 on the ground, thanks to its ingenious shape and structure.
• Civil engineering: When designing bridges, civil engineers need to consider three structural elements: beams, tension, and compression members. Using calculus, we can determine the amount of force acting on the beam including the weight of pedestrians, weight of vehicles, and the amount of traffic anticipated on that bridge. Based on these factors, the materials, size, and capacity can be computed. One famous example of a bridge that uses calculus is the Golden Gate Bridge in San Francisco. The bridge has a main span of 1,280 meters and a clearance of 67 meters above water. The bridge can withstand winds up to 160 km/h and earthquakes up to magnitude 8.3.
• Electrical engineering: Integral and differential calculus is crucial for calculating voltage or current through a capacitor. Integral calculus is also a main consideration in calculating the exact length of a power cable necessary for connecting substations that are miles apart from each other. For example, if we want to connect two substations that are 100 miles apart with a power cable that has a resistance of 0.01 ohms per mile and a capacitance of 0.0001 farads per mile, we can use integral calculus to find the length of the cable that minimizes the power loss.

Architecture and Design

Calculus is also useful for creating beautiful and functional designs in art, architecture, and graphics. Some examples are:

• Architecture: Calculus is a powerful tool that is used in architecture to help design and construct buildings and structures. Architects use calculus to calculate the rate of change of different variables, such as the slope of a roof or the curvature of an arch. This allows them to determine the optimal shape and size of different elements of a building, ensuring that it is both aesthetically pleasing and structurally sound. An example of a famous structure with calculus influenced aesthetics is the Gateway Arch in St. Louis. The main structure is a catenary, representing the shape a rope makes hanging under its own weight, and is modeled by a calculus-derived equation.
• Digital Rendering: Calculus is used for the digital rendering of light to create realistic and immersive graphics in games, movies, and other applications. One of the main challenges of rendering light is to calculate how it interacts with different surfaces and materials, such as reflection, refraction, scattering, and absorption. To do this, we need to use mathematical models that describe the physical properties of light and surfaces. Ray tracing is a technique that simulates the path of light rays as they bounce off surfaces in a scene by calculating the tangent lines at the roots of equations using calculus. Calculus also helps us create realistic effects such as shadows, ambient occlusion, global illumination, and reflections using various methods, such as shadow mapping, screen space ambient occlusion, or light field rendering.
• Animation Design: Calculus helps us create realistic and dynamic animations that simulate motion and interaction. For example, we can use calculus to find the position and orientation of an object at any given time based on its initial conditions and applied forces. One example of an animation that uses calculus is the bouncing ball, which involves finding the trajectory of a ball that bounces off the ground and walls with some elasticity.

Medicine

Calculus has many applications in medicine as well. It helps us understand how diseases spread, how drugs work, how organs function, and how medical devices operate. Some examples are:

• Epidemiology: Calculus is used to model the spread of infectious diseases and the effects of interventions such as vaccination or quarantine. For example, we can use calculus to find the basic reproduction number or the herd immunity threshold of a disease. The CDC uses a calculus-based model called the SEIR model to study the spread of COVID-19 and the impact of vaccination and social distancing measures. The SEIR model divides the population into four compartments: susceptible, exposed, infected, and recovered. The model uses differential equations to describe how people move from one compartment to another over time.
• Pharmacology: Calculus is used to study the absorption, distribution, metabolism, and excretion of drugs in the body. For example, we can use calculus to find the dosage or concentration of a drug that achieves a desired effect or avoids a toxic effect. 
• Physiology: Calculus is used to study the functions and interactions of various organs and systems in the body. For example, we can use calculus to find the blood flow or pressure in the arteries or veins, the oxygen uptake or carbon dioxide release in the lungs, or the glucose level or insulin secretion in the pancreas. One example of an organ that uses calculus is the heart, which pumps blood through the circulatory system. We can use calculus to find how much blood is pumped by each heartbeat or how much work is done by the heart in a day.

Computer Science & Data Analytics

Calculus is also essential for computer science and data analytics. It helps us develop algorithms, programs, software, and systems that can process, store, analyze, and visualize large amounts of data. Some examples are:

• Algorithms: Calculus helps us design efficient and accurate algorithms that can solve complex problems or perform specific tasks. For example, we can use calculus to find the optimal solution or approximation for a numerical problem such as finding roots, integrals, or derivatives. One example of an algorithm that uses calculus is Newton’s method, which iteratively finds better approximations for roots of equations using derivatives.
• Machine learning: Calculus helps us create and train artificial intelligence models that can learn from data and make predictions or decisions. For example, we can use calculus to find the optimal parameters or weights for a neural network that minimizes the error or maximizes the accuracy of its output. One example of a machine learning model that uses calculus is Google’s PageRank algorithm, which ranks web pages according to their relevance and importance. PageRank assigns a numerical score to each web page based on the number and quality of links pointing to it. The higher the score, the higher it is ranked in search results
• Data visualization: Calculus helps us create and interpret graphical representations of data that can reveal patterns, trends, or insights. For example, we can use calculus to find the best fit or regression line for a scatter plot of data points or to find the area or volume under a curve or surface that represents a probability distribution or a function.

Conclusion

As you can see, calculus is not only a fascinating and challenging subject but also a practical and powerful one. It has countless applications in various fields and disciplines that affect our lives in many ways. By learning calculus, you can not only improve your mathematical skills and logical thinking but also expand your horizons and opportunities. So don’t be afraid of calculus, embrace it and enjoy it! You never know when it might come in handy!