Integral Components: Understanding The Whole

The word integral has a variety of meanings, but it is most commonly associated with the field of mathematics. In mathematics, an integral is a numerical value that represents the area under the graph of a function within a given interval, or it can refer to a new function that is derived from the original function. This process of calculating an integral is known as integration, and it is a fundamental aspect of calculus. Beyond mathematics, the term integral is often used to describe something that is essential or integral to a whole. For example, one might say that violence is an integral part of human nature, implying that it is a fundamental or inherent aspect of who we are. Thus, the concept of integral extends beyond numbers and equations, finding its place in our understanding of the world and our role within it.

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Integration is a mathematical concept

Integration is used to find many useful quantities, such as areas, volumes, displacement, and other geometric properties. For example, if we have a flow rate of 1 liter per second, integration can be used to find the volume of water in a tank by adding up all the small bits of water. This is an example of a definite integral, which has actual values to calculate between. On the other hand, indefinite integrals are defined without upper and lower limits and are used for antiderivatives.

The concept of integration is based on a limiting procedure that approximates the area of a curvilinear region by dividing it into thin vertical slabs. This is related to the idea of limits in calculus, where algebra and geometry are combined. Limits help us study how points on a graph get closer to each other until their distance is almost zero.

Integration is a broad topic that is introduced in higher-level classes, such as in engineering or higher education. It is a challenging concept that requires a deep understanding of calculus and the various rules and techniques associated with integration. By learning and practicing these rules, students can become proficient in solving integration problems and applying them to real-world scenarios.

Integration by parts

$$\int_{{\,a}}^{{\,b}}{{u\,dv}} = \left. {uv} \right|_a^b - \int_{{\,a}}^{{\,b}}{{v\,du}}$$

Where $u$ and $v$ are functions of $x$, and $a$ and $b$ are the limits on $x$. The function $u$ should be chosen such that it becomes simpler when differentiated, and $v$ should be chosen such that it does not become more complicated when integrated.

To use this formula, we need to identify $u$ and $dv$, compute $du$ and $v$, and then use the formula. Computing $v$ is straightforward as it is just the integral of $dv$. One of the more complicated things about using this formula is correctly identifying both $u$ and $dv$. It won’t always be clear what the correct choices are, and occasionally, a wrong choice will be made. This is not something to worry about, as we can always go back and try a different set of choices.

$$\int x\cos(x)\,dx = x\sin(x) - \int \sin(x)\,dx = x\sin(x) + \cos(x) + C$$

Where integration by parts is performed twice.

Riemann integral

The Riemann integral, created by Bernhard Riemann, is a concept in the branch of mathematics known as real analysis. It was first presented to the faculty at the University of Göttingen in 1854 and later published in a journal in 1868. The Riemann integral is the first rigorous definition of the integral of a function on an interval.

The Riemann integral can be evaluated using the fundamental theorem of calculus, approximated by numerical integration, or simulated using Monte Carlo integration. It involves finding the area under a curve on a graph between two points, often denoted as "a" and "b". This area can be described as the set of all points (x, y) on the graph that satisfy the conditions: a ≤ x ≤ b (the x-coordinate is between a and b) and 0 < y < f(x) (the y-coordinate is between 0 and the height of the curve f(x)).

The Riemann integral is defined in terms of Riemann sums. The area under a function is approximated as a sum of rectangles, and as the width of these rectangles gets smaller, the approximation improves. The sum of the areas of these rectangles converges to a number, which is defined as the Riemann integral of the function. This process is similar to the Darboux integral, which is based on Darboux sums. If a function is continuous, the lower and upper Darboux sums for an untagged partition are equal to the Riemann sum for that partition.

A function is considered Riemann-integrable if there are two step functions, one above and one below it, and the area between their graphs can always be made smaller than any given quantity. This means that the area between the curve and the x-axis can be approximated arbitrarily closely using step functions. If a function is Riemann-integrable, the value of the Riemann integral is the limit of the sums as the partitions become finer.

The Riemann integral is a fundamental concept in calculus and is widely used in physics and engineering. It provides a method to approximate the integral of a function and has been a significant contribution to the field of real analysis.

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Lebesgue integral

The Lebesgue integral is an alternative way of defining the integral in terms of measure theory. It is a generalisation of the Riemann integral, which considers the area under a curve as made out of vertical rectangles. In contrast, the Lebesgue integral considers horizontal slabs that are not necessarily just rectangles, making it more flexible and able to calculate integrals for a broader class of functions. This approach to integration was summarised by Lebesgue himself in a letter to Paul Montel:

> "I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral."

The key insight behind the Lebesgue integral is that one should be able to rearrange the values of a function freely while preserving the value of the integral. This process of rearrangement can convert a pathological function into one that is easier to integrate. Specifically, the Lebesgue integral defines the integral in a way that does not depend on the structure of \(\mathbb{R}\), allowing it to integrate many functions that cannot be integrated using the Riemann integral.

The Lebesgue integral also addresses some of the limitations of the Riemann integral. For example, the Riemann integral does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyse. The Lebesgue integral, on the other hand, describes better how and when it is possible to take limits under the integral sign, thanks to theorems like the monotone convergence theorem and dominated convergence theorem.

Calculus

The definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. The fundamental theorem of calculus requires the lower limit to be a constant and the upper limit to be a variable. The number "a" at the bottom of the integral sign is called the lower limit of the integral, and the number "b" at the top is called the upper limit.

Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve or determining displacement from velocity. The geometric significance of the integral is that it gives a measure of the area enclosed by the graph of the function. Integration was first rigorously formalized using limits by Riemann. However, more general functions were considered, particularly in the context of Fourier analysis, to which Riemann's definition does not apply.

In engineering, professionals encounter integral calculus in various practical applications, such as speed modelling in highway engineering.

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