What is Integral: Definition and 1000 Discussions

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
The integrals enumerated here are those termed definite integrals, which can be interpreted formally as 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. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

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  1. B

    A Help needed with derivation: solving a complex double integral

    I need help with a derivation of an equation given in a journal paper. My question is related to the third paragraph of this paper: https://doi.org/10.1007/BF00619826. Although it is about fibre coupling my problem is purely mathematical. It is about solving a complex double integral. The...
  2. K

    I Approximating discrete sum by integral

    I can't understand how this approximation works ##\sum_{k=0}^m\left(\frac{k}{m}\right)^n\approx\int_0^m\left(\frac{x}{m}\right)^ndx\tag{1}##Can you please help me
  3. wrobel

    I The behavior of a potential-like integral at infinity

    I need a help in the following problem. I feel that the question is stupid. Take a function ##f\in C(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)## and a number ##\alpha\in(0,3)##. Prove that $$\lim_{|x|\to\infty}\int_{\mathbb{R}^3}\frac{f(y)dy}{|x-y|^\alpha}=0.$$ I can prove this fact by the Uniform...
  4. Addez123

    What are the limits for integrating a constrained surface with two variables?

    I start by parametarize the surface with two variables: $$r(u,v) = (u, v, \frac {d -au -bv} c)$$ The I can get the normal vector by $$dr/du \times dr/dv$$ What limits should I use to integrate this only within the elipse? I could redo the whole thing and try write r(u, v) as u being the...
  5. wrobel

    I A problem with non evident first integral

    The following problem we considered with the students. Perhaps it would also be interesting for PF A homogeneous rod can rotate freely in a plane about its (fixed) center of mass O . The corresponding moment of inertia is equal to J. Two identical particles of mass m can slide along the rod...
  6. Rlwe

    I Is the sign of the integral of this function negative?

    Let ##f:[0;1)\to\mathbb{R}## and ##f\in C^1([0;1))## and ##\lim_{x\to1^-}f(x)=+\infty## and ##\forall_{x\in[0;1)}-\infty<f(x)<+\infty##. Define $$A:=\int_0^1f(x)\, dx\,.$$ Assuming ##A## exists and is finite, is it possible that ##\text{sgn}(A)=-1##?
  7. A

    U-Substitution in trig integral

    ##\int \frac{\csc{x}\cot{x}}{1+\csc^2{x}}dx## Let ##u = \csc{x}## then ##-du = \csc{x}\cot{x}dx## So, ##\int \frac{\csc{x}\cot{x}}{1+\csc^2{x}}dx## ##-\int \frac{1}{1+u^2}du = -\arctan{u} + C## ##-\arctan{\csc{x}} + C## This answer was wrong. The actual answer involved fully simplifying...
  8. A

    Question about a double integral region

    Greetings All! I have a problem finding the correct solution at first glance My error was to determine the region of integration , for doing so I had to the intersection between y= sqrt(x) and y=2-x to do so x=(2-x)^2 to find at the end that x=1 or x=5 while graphically we can see that the...
  9. A

    Problem with the region of an integral

    Greetings! The exercice ask to calculate the circuitation of the the vector field F on the border of the set omega I do understand the solution very well my problem is the region! I m used to work with a region delimitated clearly by two intersecting function here the upper one stop a y=3 and...
  10. G

    I Can a Path Integral Formulation for Photons Start from a Massless Premise?

    I am aware that one usually starts from the Maxwell equations and then derives the masslessness of a photon. But can one do it the other way round? The action of photon would be ##S = \hbar \int \nu (1 - \dot{x}^2) \mbox{d}t##, where ##\nu## is the frequency acting as a Lagrange multiplier...
  11. A

    A Solving an Integral involving a probability density function

    In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows: "For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...
  12. T

    A Help with simplifying an integral

    I am not seeing how the v goes away in the third equal sign of equation (1.8). It seems to be that it must be cos(z*sinh(u+v)), not cos(z*sinh(u)). In the defined equations (1.7), the variable "v" can become imaginary, so a simple change of variables would change the integration sign by adding...
  13. Lorena_Santoro

    MHB Is It an $\arctan$, $\arccot$, or Something Else in This Integral Solution?

  14. rudransh verma

    B What is the concept of distance as an integral in Feynman lectures on physics?

    From Feynman lectures on physics: https://www.feynmanlectures.caltech.edu/I_08.html Page 8-7 (Ch 8 Motion) “ To be more precise, it is the sum of the velocity at a certain time, let us say the ith time, multiplied by deltat. ##s=\sum_{i} v({t_i})\Delta t##” Now I suppose ##{t_i}## is some time...
  15. Vividly

    I Why is the absolute value of sinx used in the integral of cotx?

    Why is there a absolute value sign on sinx? Does it have to do with the domain of cot x and sin x?
  16. Lorena_Santoro

    MHB You have only 3 minutes for a better solution to this integral

  17. Lorena_Santoro

    MHB What's the best strategy to solving this Integral in 3 minutes?

  18. docnet

    What is the solution to the nested integral problem?

    Let ##u=\int_1^{u}2xdx##. \begin{align}u=& \int_1^{u}2xdx=\big[x^2\big]_1^u\\ u=&u^2-1\end{align} Which leads to ##u=\frac{1\pm\sqrt{1+4}}{2}## Assuming that the upper boundary of integration is greater than ##1##, or less than ##-1##, leads to ##u=\frac{1+\sqrt{5}}{2}\approx 1.61##. the second...
  19. S

    Integral using substitution x = -u

    Is it possible to solve this integral? I think the substitution ##x=-u## does not help at all since it only changes variable ##x## to ##u## without changing the integrand much. Using that substitution: $$\int \frac{6x^2+5}{1+2^x}dx=-\int \frac{6u^2+5}{1+2^{-u}}du$$ Then how to continue? Thanks
  20. jaumzaum

    I Why does the integral of sine of x^2 from - infinity to + infinity diverge?

    Hello guys. I was trying to evaluate the integral of sine of x^2 from - infinity to + infinity and ran into some inconsistencies. I know this integral converges to sqrt(pi/2). Can someone help me to figure out why I am getting a divergent answer? $$ I = \int_{-\infty}^{+\infty} sin(x^2) dx =...
  21. docnet

    Compute the integral of the Gaussian

    why does it say transforms? is there more than one Fourier transform?? we learned in class that the inverse Fourier transform of the Fourier transform of ##f## is ##f##, so there should be just one right? I'm uncertain of how to calulate this integral though.. Mr Wolfram showed me an indefinite...
  22. Tapias5000

    Integral ## \int _{ }^{ }\frac{1}{\sqrt{x^3+1}}dx ##

    I also don't understand how to get the descending factorials for this hypergeometric series, I also know that there is another way to write it with gamma functions, but in any case how am I supposed to do this? If I write it as a general term, wolfram will give me the result which leaves me...
  23. C

    Prove limit comparison test for Integrals

    Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
  24. A

    A How to solve this definite integral?

    I would like to solve the integral underneath: $$\displaystyle \int_{0}^{x}\!-{\frac {\lambda\,{{\rm e}^{-\lambda\,t}}{\beta}^{\alpha} \left( -\lambda+\beta \right) ^{-\alpha} \left( -\Gamma \left( \alpha \right) +\Gamma \left( \alpha, \left( -\lambda+\beta \right) t \right) \right)}{\Gamma...
  25. Addez123

    Simple integral, can't get the right answer....

    $$\int \frac y {x^2+y^2} dx$$ $$\frac 1 y * \int \frac 1 {\frac {x^2}{y^2} + 1} dx = \frac 1 y * atan(x/y)$$ The answer is just atan(x/y), which you get using u-substitution but I honestly don't see why I don't get it doing it the normal way.
  26. brotherbobby

    A line integral about a closed "unit triangle" around the origin

    Problem statement : As a part of the problem, the diagram shows the contour ##C##above on the left. The contour ##C## is divided into three parts, ##C_1, C_2, C_3## which make up the sides of the right triangle. Required to prove : ##\boxed{\oint_C x^2 y \mathrm{d} s = -\frac{\sqrt{2}}{12}}##...
  27. Wizard

    A Orthogonality of variations in Faddev-Popov method for path integral

    Hi there, I've been stuck on this issue for two days. I'm hoping someone knowledgeable can explain. I'm working through the construction of the quantum path integral for the free electrodynamic theory. I've been following a text by Fujikawa ("Path Integrals and Quantum Anomalies") and also...
  28. Johan_S

    A Yet another cross-product integral

    I am trying to integrate a cross product and I wonder if the following is true. It does not feel like it is true but it would be very nice if it was since otherwise I have a problem with the signs... This is my first time posting here, so I just pasted in the LaTeX code and hope that it is...
  29. brotherbobby

    Line integral of a scalar function about a quadrant

    Problem : We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##. Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using (plane) polar coordinates. (1) In (plane)...
  30. J

    The integral does not converge....

    I am asked to compute ##[\phi(x), \phi^\dagger(y)]## , with ##\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})## and with z=x-y a spacelike vector. And show that this commutator does not vanish, which means that for this non-relativsitic field i.e. with ##p^0 = \frac{\vec{p}^2}{2m}##...
  31. karush

    MHB -7.3.89 Integral with trig subst

    $\begin{array}{lll} I&=\displaystyle\int{\frac{dx}{x^2\sqrt{x^2-16}}} \quad x=4\sec\theta \quad dx=4\tan \theta\sec \theta \end{array}$ just seeing if I started with the right x and dx or is there better Mahalo
  32. Einstein44

    Question about the Magnetic Flux equation in Integral form

    .
  33. F

    "Trick" for a specific potential function defined with an integral

    Hello, To first clarify what I want to know : I read the answer proposed from the solution manual and I understand it. What I want to understand is how they came up with the solution, and if there is a way to get better at this. I have to show that, given a vector field ##F## such that ## F ...
  34. derya

    A Analytical solution for an integral in polar coordinates?

    Hi, I am trying to find open-form solutions to the integrals attached below. Lambda and Eta are positive, known constants, smaller than 10 (if it helps). I would appreciate any help! Thank you!
  35. J

    Calculators TI 89 Integral seems wrong.... What am I missing?

    My TI 89 Platinum is returning ln(abs(cos(x))/abs(sin(x)-1)) for integral sec(x)dx. It's supposed to return ln(abs(tan(x)+sec(x)) or ln(abs(sin(x)+1)/abs(cos(x))). If you enter x=0, you get 'undefined' the way my TI 89 is doing it. It's supposed to return 0. Is this a computation error or...
  36. A

    Integral of 1 / (x^2 + 2) dx ?

    Mentor note: Moved from technical section, so missing the homework template. How do you integrate this? $$\int \frac{1}{x^2 + 2} dx$$ My attempt is $$\ln |x^2 + 2| + C$$
  37. MathematicalPhysicist

    I A computation of an integral on page 344 of Schutz's textbook

    On page 344 of "A First Course in GR" he writes the following: When I do the integration I get the following: ##\int_0^{\chi^2}d\chi^2= \int_0^{r^2}\frac{dr^2}{1-r^2}= \chi^2 = -\ln (1-r^2)##, after I invert the last relation I get: ##r=\sqrt{1-\exp(-\chi^2)}##, where did I go wrong in my...
  38. O

    Arc length of vector function - the integral seems impossible

    The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...
  39. K

    I Feynman path integral and events beyond the speed of light

    In Richard Feynman's book "The Strange Theory of Light and Matter", in chapter 2, he explains how to calculate the probability that light from some source will be reflected by a mirror and be detected at some location. He explains how you sum up all of the probability amplitudes (represented...
  40. raisins

    I Phase space integral in noninteracting dipole system

    Hi all, Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is...
  41. E

    MHB Indefinite integral in division form

    I have the following integration - $$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx $$ To solve this I did the following - $$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$ Which gives me -...
  42. docnet

    B Are there mathematicians that dislike integral calculus?

    Solving integrals by hand is difficult and prone to errors, and the techniques such as integration by parts, partial fraction decomposition, and trig substitutions only work for a small subset of integrals and I do not see the point of avoiding technology like Wolfram Mathematica for...
  43. M

    MHB Indeterminate Integration with Integration Constant

    Hey! 😊 I want to calculate the integral $$\int\frac{1}{(x+4)(x^2-8x+19)}\, dx$$ I have done the following : $$\frac{1}{(x+4)(x^2-8x+19)}=\frac{1}{67}\frac{1}{x+4}+\frac{1}{67}\frac{12-x}{x^2-8x+19}$$ and so we get \begin{align*}\int\frac{1}{(x+4)(x^2-8x+19)}\, dx&=\frac{1}{67}\int...
  44. A

    Why this triple integral is not null?

    Greetings here is my integral Compute the volume of the solid and here is the solution (that I don't agree with) So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric...
  45. A

    Double integral with polar coordinates

    Greetings! I have the following integral and here is the solution of the book (which I understand perfectly) I have an altenative method I want to apply that does not seems to gives me the final resultMy method which doesn't give me the final results! where is my mistake? thank you!
  46. Safinaz

    B How to make this integral with initial conditions

    Hello! The integral in equation (16), at the paper, is: ##I = r \int_{-\pi}^{\pi} e^{-2kr\phi} ~d\phi ## My integration is as the following : ## I = - \frac{1}{2 k} e^{-2kr\phi} ~|_{-\pi}^{\pi} + C ##, so ## I = - \frac{1}{2 k} ( e^{-2kr\pi} -e^{2kr\pi})+ C ## Now how to use the initial...
  47. Istiak

    How integral and gradient cancels?

    I know that gradient is multi-variable derivatives. But, here line integration (one dimensional integral) had canceled gradient. How?
  48. H

    Overlap integral of hydrogen molecule

    Hi! Some help with this problem would be much appreciated. The overlap integral is defined as ##S = \int \phi_A (\mathbf{r}_A) \phi_B (\mathbf{r}_B) \,d\mathbf{r}##. For the two orbitals, I have that $$\phi_A = \frac{1}{\sqrt{\pi}} \Big( \frac{1}{a_0} \Big)^{3/2} e^{-r_A / a_0}$$ for the 1s...
  49. DaalChawal

    MHB Question from definite Integral

  50. newjerseyrunner

    I Creating a function with specific shape, intercepts, integral....

    I'm trying to see if I can calculate the peak draw weight of my bow based on the draw length and the velocity of the arrow and a known shape of a curve, but I'm not quite sure how to make such a function, if there even is such a way. This is the shape of the draw weight plotted against...
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