Integral

In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Over time, its use expanded to many scientific fields.

A definite integral computes the signed area of a region bounded by the graph of a function between two points on the real line. Areas above a certain line are considered positive, while areas below are negative. Integrals also relate to the idea of an antiderivative, a function whose derivative gives the original function; these are called indefinite integrals. The fundamental theorem of calculus connects integration with differentiation and offers a way to compute integrals when an antiderivative is known.

The idea of calculating areas and volumes dates back to ancient Greek mathematics, but the principles of integration were developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. They thought of the area under a curve as an infinite sum of very thin rectangles. Later, Bernhard Riemann provided a precise definition of integrals using a method that breaks regions into thin slabs. In the early 20th century, Henri Lebesgue introduced a more general type of integral now known as the Lebesgue integral.

Integrals can be adapted for different situations. For example, a line integral applies to functions with two or more variables, where the integration path is a curve in space. In a surface integral, the path becomes a piece of a surface in three-dimensional space. These tools help solve complex problems in many areas of science and engineering.

History

See also: History of calculus

The history of integration began long ago with ancient mathematicians trying to find areas and volumes. The ancient Greek astronomer Eudoxus and philosopher Democritus developed a method called the method of exhaustion around 370 BC. This method involved breaking shapes into many small parts to find their area or volume. Later, Archimedes used this method to calculate the area of a circle, the surface area and volume of a sphere, and other shapes like the area under a parabola.

Similar ideas were also used in China around the 3rd century AD by Liu Hui, and later by Zu Chongzhi and Zu Geng. In the Middle East, Alhazen made advances in calculating areas and volumes using early integral methods.

The big breakthrough came in the 17th century with Leibniz and Newton, who discovered the fundamental theorem of calculus. This theorem showed a deep connection between integration and differentiation, making it much easier to solve many problems. They developed a precise mathematical framework that became modern calculus. The symbol ∫ for integration was introduced by Leibniz in 1675, adapted from an old form of the letter S, standing for "sum."

Terminology and notation

The integral of a function is written using a special symbol: ∫. This symbol represents the process of integration. When we write ∫ from a to b of f(x) d_x, we are talking about a definite integral. Here, f(x) is the function we are integrating, and a and b are the limits that tell us the range we are focusing on.

If we see ∫ f(x) d_x without limits, it is called an indefinite integral. This represents a group of functions whose rate of change (derivative) is the original function. The connection between definite and indefinite integrals is explained by the fundamental theorem of calculus.

Interpretations

Integrals help us solve many real-world problems. For example, they can calculate the volume of water in swimming pools, even if the pool has an unusual shape like an oval with a rounded bottom. To find these exact values, we imagine dividing the pool into tiny, tiny pieces and then add up all those pieces.

We can also use integrals to find the area under a curve, like the shape made by the function f(x) = √x between x = 0 and x = 1. By dividing this area into small sections and adding up their areas, we can get very close to the true area. As we use more and more sections, getting closer and closer together, we reach the exact area, which in this case is 2/3.

Formal definitions

There are many ways to formally define an integral, but the most common are Riemann integrals and Lebesgue integrals. The Riemann integral is defined using something called "Riemann sums," which break an area into small rectangles and add up their areas. This method works well for many everyday shapes and curves.

The Lebesgue integral is another way to define integrals that can handle more complicated situations. It looks at how the values of a function are spread out and adds up areas in a different way. Both types of integrals help us understand and calculate areas under curves, which is a key idea in calculus.

Fundamental theorem of calculus

Main article: Fundamental theorem of calculus

The fundamental theorem of calculus tells us that finding the area under a curve and finding the slope of a curve are opposite operations. If we first find the area under a curve and then find the slope of that area, we get back to the original curve. This important idea helps us compute integrals using something called an antiderivative, which is a special kind of function related to the original one.

Extensions

Improper integrals

Main article: Improper integral

An improper integral is when the usual rules for integration don’t apply, like when the area goes on forever or the value at a point isn’t defined. We can still find these integrals by looking at limits. For example, if we’re trying to find the area under a curve that goes on forever, we look at what happens as we go further and further out.

Multiple integration

Main article: Multiple integral

Just like how we can find the area under a curve, we can also find the volume under a surface. For example, if we have a shape in 3D space, we can use a double integral to find its volume. This works by breaking the shape into very thin slices and adding up their volumes.

Line integrals and surface integrals

Main articles: Line integral and Surface integral

We can also use integrals to find things like the work done by a force along a path, or the flow of a fluid through a surface. For example, if a toy moves through an invisible force field, we can use a line integral to find the total effect of that force on the toy.

Contour integrals

Main article: Contour integration

In complex numbers, we can also integrate along paths in the complex plane. This is useful in solving many problems in math and physics.

Integrals of differential forms

Main article: Differential form

See also: Volume form and Density on a manifold

Differential forms are a way to generalize integrals to more complicated shapes and spaces. They help us understand how things change in higher dimensions and on curved surfaces.

Summations

Main article: Summation § Approximation by definite integrals

Summations are like adding up a list of numbers, while integrals are like adding up infinitely many tiny pieces. They are closely related and can sometimes be used in place of each other.

Functional integrals

Main article: Functional integration

Sometimes we want to integrate over all possible functions, not just numbers or points in space. This is called a functional integral and is used in advanced physics and math.

Applications

Integrals are important in many areas. In probability theory, they help find the chance that a random variable falls within a certain range. They are also used to find the area of shapes with curved edges and the volume of three-dimensional objects, like disks, by using special formulas.

Integrals are useful in physics, such as in kinematics, to determine how far an object has moved over time. They are also used in thermodynamics to calculate energy differences between states.

Computation

Main article: Symbolic integration

Main article: Numerical integration

Main article: Quadrature (mathematics)

In math, integration helps us find the area under a curve or the total amount of something by adding up many small pieces. There are different ways to do this.

One way is called analytical integration. We look for a special kind of function called an "antiderivative" that, when we do certain math to it, gives us back the original function. If we can find this antiderivative, we use it to calculate the exact value of the integral.

Another way is numerical integration. Here, we don’t find the exact answer, but we get very close by breaking the area into small pieces, like rectangles or trapezoids, and adding up their areas. This gives us a good estimate of the total area.

Examples

The fundamental theorem of calculus helps us easily calculate integrals for basic functions. For example, we can find the integral of the sine function from 0 to π, which equals 2. This shows how integration can give us useful results in mathematics.

Main article: fundamental theorem of calculus