Integral

In mathematics, an integral is like a continuous version of adding up things. We use it to find areas, volumes, and more. Finding an integral is called integration. It is one of the two big ideas in calculus, the other being differentiation. People first used integration to solve problems in math and physics, like finding the space between two points on a line.

A definite integral helps us find the exact area under a curve between two points. Areas above the line count as positive, and areas below count as negative. Integrals also connect to something called an antiderivative, which is a function that, when changed, gives the original function. These are called indefinite integrals. The fundamental theorem of calculus shows how integration and differentiation are linked and helps us calculate integrals when we know an antiderivative.

The idea of finding areas and volumes started with ancient Greek mathematics. The rules for integration were later created by Isaac Newton and Gottfried Wilhelm Leibniz in the late 1600s. They thought of the area under a curve as adding up infinitely many thin rectangles. Then, Bernhard Riemann gave a clear way to define integrals. In the early 1900s, Henri Lebesgue created a more general kind of integral called the Lebesgue integral.

Integrals can be changed to fit different situations. For instance, a line integral is used for functions with more than one variable, where we integrate along a curve. In a surface integral, we integrate over a piece of a surface in three-dimensional space. These tools help solve tough problems in 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 broke 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. They developed a precise mathematical framework that became modern calculus. The symbol ∫ for integration was introduced by Leibniz in 1675. It came 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 shows 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 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, we reach the exact area, which in this case is 2/3.

Formal definitions

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

The Lebesgue integral is another way to define integrals. It 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. This is a key idea in calculus.

Fundamental theorem of calculus

Main article: Fundamental theorem of calculus

The fundamental theorem of calculus shows us that finding the area under a curve and finding the slope of a curve are opposite actions. If we first find the area under a curve and then find the slope of that area, we return to the original curve. This important idea helps us calculate integrals using something called an antiderivative, a special kind of function related to the original one.

Extensions

Improper integrals

Main article: Improper integral

An improper integral is when the normal rules for integration don’t work, like when the area goes on forever or a value is missing. We can still find these integrals by looking at limits. For example, if we want the area under a curve that never ends, we watch what happens as we go really far out.

Multiple integration

Main article: Multiple integral

Just like 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 cutting 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 use integrals to find things like the work done by a force along a path, or how much fluid moves 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 helps solve 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 complex 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 instead 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 is within a certain range. They are also used to find the area of shapes with curved edges and the volume of objects like disks.

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

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." 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 find integrals for basic functions easily. For example, we can find the integral of the sine function from 0 to π, and the result is 2. This shows how integration can give us useful results in mathematics.

Main article: fundamental theorem of calculus