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Calculus is a branch of mathematics divided into differential calculus and integral calculus. It is used to study the motion or change experienced by certain variables.
This is the essence of calculus and is fundamental in a world that is constantly changing, since the only constant we have in the universe is precisely change.
Calculus, as a branch of mathematics, originated in ancient Greece with the purpose of estimating the magnitude of changes in variables and determining lengths, areas, volumes, among others.
This area of mathematics evolved into a modern discipline, attributing new contributions to mathematicians Isaac Newton and Gottfried Leibniz, who are credited with the fundamental theorem of calculus, specifically infinitesimal calculus, dedicated to the study of limits, derivatives, infinite series, and integrals, the latter being called integral calculus.
Calculus, also called infinitesimal calculus, is the branch of mathematics that focuses on the study of continuous change.
It is also a branch of mathematical analysis that studies how continuous functions change as their variables change state. The main object of study in differential calculus is the derivative.
The study of the change of a function is of special interest to differential calculus, specifically the case in which the change of variables is infinitesimal, that is, when the change tends to zero (it becomes as small as desired).
Calculus allows us to study, among other things, location, volume, area, and velocity.
Studying the position of things: In terms of location, there is no problem determining the location of something that is fixed; but what happens when changes in position occur?
For example, when an object is moving perpendicular to another object. This represents a certain analytical difficulty that has answers thanks to differential calculus.
Studying volume: Assuming we have a sphere whose dimensions are constantly changing, how could we analyze its volume without major problems? With calculus.
Studying Area: It's easy to determine the area occupied by an object placed on a table surface, but what if that object is expanding?
Using calculus, we could analyze this expanding surface.
Studying Speed: It can be easy to determine the speed of an object when its motion remains at a constant speed.
But, for example, calculus can be used to determine the speed of a car traveling from one city to another, taking into account traffic lights, jams, and other factors that cause the speed to constantly vary.
Integral calculus is a branch of calculus that focuses on the study of integrals and their applications. An integral is a mathematical concept that represents the area under a curve in a given interval.
Integral calculus is used to solve problems involving continuously varying quantities, such as the velocity of a moving object, the area under a curve, the volume of a three-dimensional solid, among others.
Integral calculus is based on the concept of limits and the sum of infinitely many infinitesimal elements. It is used to find areas, volumes, arc lengths, centers of mass, moments of inertia, among others. In addition, it has applications in diverse areas such as physics, engineering, economics, and biology, among others.
Símbolo de la integral: La integral se denota comúnmente con el símbolo ∫, que proviene del latín sumatoria. La notación completa incluye la función a integrar, los límites de integración y el elemento diferencial, por ejemplo:
Function to be integrated: The function being integrated is the central element of the integral. It can be a continuous or discontinuous function on the integration interval.
Difference element: This is the differential of the integration variable. In a definite integral, the function is multiplied by this differential element. It is commonly denoted as dx in an integral with respect to x, dy in an integral with respect to y, etc.
These are the elements that make up an indefinite integral, additionally, an indefinite integral which will be a more advanced topic of integral calculus, also includes the limits of integration.
When solving an integral we will obtain the primitive function, also called antiderivative, that is, given a function f(x) we must find its primitive F(x) whose derivative is equal to f(x);
It should be noted that there are some functions f(x) that satisfy this relationship, for this reason it is said that the integral is indefinite, but if certain conditions are registered for F'(x) where it becomes unique, the integral is considered a definite integral, said integral being interpreted geometrically as the area between the X axis, the graph of the function or curve and the ordinates delimited by two values of X (b,a) called upper limit and lower limit respectively.
The integrals form presents us with a list of integrals with their respective results, some of them are immediate integrals, that is, integrals that do not require applying any integration method because they are very simple. This form makes it easier to solve exercises where the functions have a complex structure.
The most commonly used formulas are the following:
Click here to download the formula table.
There are two fundamental theorems that we must know and understand when studying integral calculus:
Fundamental Theorem of Calculus: It intuitively consists of the statement that the differentiation and integration of a function are inverse operations.
Mean Value Theorem States that if a function is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c contained in the interval (a,b) such that f'(c) is equal to the average rate of change of the function on [a,b].
In addition to definite and indefinite integrals, there are others worth studying, such as:
Improper integrals: These are integrals that present a vertical asymptote in the integration interval, or whose integration interval is not limited.
Line integrals: These are integrals whose function is evaluated on a curve.
Trigonometric integrals: These are integrals whose integral is composed of trigonometric functions and constants, including the sine integral, cosine integral, tangent integral, and secant integral.
Multiple integrals: These include double integrals and triple integrals.
Iterated integral: This is an integral evaluated several times on the same variable.
Differential calculus is a branch of mathematics that has played a crucial role in understanding and describing a wide range of phenomena in science, engineering, and beyond.
Differential calculus focuses on the study of rates of change and how functions vary instantaneously at specific points. Its main focus is the concept of the derivative, which provides a precise measure of the rate at which one quantity changes relative to another.
Differential calculus allows us to understand how functions behave when their infinitesimal change in one variable is analyzed. It becomes an invaluable tool in modeling and solving real-world problems, from physics and economics to biology.
The derivative, a fundamental element of differential calculus, gives us instant insight into how a function evolves at a specific point. Furthermore, the geometric interpretation of the derivative as the slope of a line tangent to a curve reflects the deep connection between calculus and geometry.
The derivative of a function is understood to be the instantaneous rate of change at which the value of said function varies according to the value of its independent variable. Therefore, we speak of the value of the derivative of a function at a given point.
The derivative of a function, from the point of view of geometry, is nothing more than the slope of the tangent line of the function f(x). Therefore, it is defined by taking the limit of the slope of the secant lines when bringing them closer to the tangent line.
By definition, it is said that the derivative of a function y=f(x) with respect to x at a point (a) is:
Expressed another way, the derivative of a function is the limit between the increase in the dependent variable and the increase in the independent variable as it tends to zero
The derivative of y=f(x)> with respect to «x» > can be denoted in several ways:
d/dx y; dy/dx; y´; f´(x); d/dx f(x)
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The fields of application of this discipline are very broad, as its usefulness extends to a wide variety of situations, such as medicine, astronomy, physics, engineering, statistics, computing, business, and demography .
Through calculus, very important answers can be found in each field, for example:
In medicine, it reveals the trajectory of medications over time.
Astronomy uses it to direct satellites toward nearby asteroids.
Physics uses it to accurately measure distances, speeds, and accelerations.
Statistics uses it to accurately define probability distributions.
In computing, it is deployed to analyze animations in detail.
In business, it calculates costs after producing x units.
Demography uses it to identify changes in population dynamics.
These are just simple examples that illustrate the wide scope and usefulness of calculus, and there are many more like these. Calculus is present in virtually every aspect of our lives.
Remember to review the answers to the open-ended questions at the bottom of this page.
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We recommend you visit the following material for greater knowledge or understanding on the topic:
1. Calculus: Britannica 2. Calculus7. In the study of integrals and their applications.
8. Integral symbol, function to be integrated, and difference element.
9. Improper, linear, trigonometric, multiple, and iterated derivatives.
10. Derivatives
References:
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