What is Integrals: 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. O

    I How Accurate is Propositional Logic in Explaining Multiple Integrals?

    The following 3 pages are extract from the book: "CALCULUS VOL II" by Tom M. Apostol My interpretation of these 3 pages is worked out in the attached PDF file. Entirely done in propositional logic language. Can anyone point out the mistakes or incorrect logical steps (if any) in the attached...
  2. A

    MHB Good day, Exam Integrals: volume and area

    1#Find the area of the region, enclosed by: 2#Find the area of the region bounded by: 3#in the region limited by: find the solid volume of revolution that is generated by rotating that region about the x axis
  3. A

    Where to use polar (cylindrical coor.) in double and triple integrals

    where the region of integration is the cube [0,1]x[0,1]x[0,1] my question is where can we use the polar coordinate? is it only usable if the region of integration looks like a circle regardless of the function inside the integral? (if yes it means that using this kind of transformation is wrong...
  4. I

    I Phase space path integrals

    I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...
  5. MountEvariste

    MHB What is the ratio of two integrals involving sine with exponents of sqrt(2)?

    $ \displaystyle I = \int_0^{\pi/2} \sin^{\sqrt{2}+1}{x}$ and $\displaystyle J = \int_0^{\pi/2} \sin^{\sqrt{2}-1}{x}$. Find $\displaystyle \frac{I}{J}.$
  6. D

    Using Surface Integrals, calculate the area that vanishes with this rising tide

    Please help to see whether it's correct to do in this way
  7. B

    A Jacobi Elliptic Functions and Integrals

    Are there any useful references or resources that intuitively show how Jacobi Elliptic functions [sn, cn, dn, etc] are geometrically interpreted from properties of ellipses? And how the Jacobi Elliptic functions and integrals can be shown to be generalizations of circular trig functions? Thanks!
  8. Adesh

    I Why integrals took 2000 years to come up in a rigorous manner?

    Archimedes Riemann integral is one of the most elegant achievements in mathematics, I have a great admiration for it. Mr. Patrick Fitzpatrick commented on it as Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer...
  9. S

    A Fractional Calculus - Variable order derivatives and integrals

    Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...
  10. Adesh

    How to prove that ##M_i =x_i## in this upper Darboux sum problem?

    We're given a function which is defined as : $$ f:[0,1] \mapsto \mathbb R\\ f(x)= \begin{cases} x& \text{if x is rational} \\ 0 & \text{if x is irrational} \\ \end{cases} $$ Let ##M_i = sup \{f(x) : x \in [x_{i-1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 ...
  11. A

    I Equality of integrals VS equality of integrands

    Does $$\int_{t=0}^{\infty}f(t)dt=\int_{t=0}^{\infty}g(t)dt$$ imply $$f(t)=g(t)$$ ?
  12. fresh_42

    Applied Integrals, Series and Products

    I recently stumbled upon Gradshteyn, Ryzhik: Table of Integrals, Series, and Products and it is worth recommending for all who have to deal with actual solutions, i.e. especially engineers, physicists and all who are confronted with calculating integrals, series and products.
  13. Adesh

    I Understanding the ##\epsilon## definition of this integral

    Integrals are defined with the help of upper and lower sums, and more number of points in a partition of a given interval (on which we are integrating) ensure a lower upper sum and a higher lower sum. Keeping in mind these two things, I find the following definition easy to digest A function...
  14. B

    Calculus 1 problems: functions, integrals, series

    Mentor note: Moved from technical section, so is missing the homework template. Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?
  15. M

    Mathematica Running a matrix of integrals in parallel

    Hi PF! I am trying to computer a matrix of integrals. Think of it something like this: Table[Integrate[x^(i*j), {x, 0, 1}], {i, 0, 5}, {j, 0, 5}] I have 16 cores, and would like to have each core handle a specified amount of integrals. Anyone know how to do this? Thanks so much!
  16. A

    A Algebra of divergent integrals

    Hello, guys! I would like to know your opinion and discuss this extension of real numbers: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 In essence, it extends real numbers with entities that correspond to divergent integrals and series. By adding the rules...
  17. A

    I Average of the B-field over a volume and surface integrals

    Purcell says that taking the surface integral of the magnetic field ##\textbf{B}## over the surfaces ##S_{1}, S_{2}, S_{3},...## below is a good way of finding the average of the volume integral of ##\textbf{B}## in the neighborhood of these surfaces. More specifically, he says in page...
  18. O

    I Is this PDF file the correct derivation?

    My work is in the following pdf file:
  19. VVS2000

    Surface integrals to calculate the area of this figure

    I can find the area of the triangles but can't solve the squares for some reason
  20. Hiero

    I Optimization of multiple integrals

    The Euler Lagrange equation finds functions ##x_i(t)## which optimizes the definite integral ##\int L(x_i(t),\dot x_i(t))dt## Is there any extensions of this to multiple integrals? How do we optimize ##\int \int \int L(x(t,u,v),\dot x(t,u,v))dtdudv## ? In particular I was curious to try to...
  21. A

    I Changing variables in multiple integrals

    Suppose we have a region R in the x-y plane and divide the region into small rectangles of area dxdy. If the integrand or the limits of integration were to be simplified with the introduction of new variables u and v instead of x and y, how can I supply the area element in the u-v system in the...
  22. W

    I Propagator of a Scalar Field via Path Integrals

    I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...
  23. P

    A Polylogarithms integrals in Nastase QFT book

    This is from Horatio Nastase "Intro to Quantum Field Theory" book (Cambridge University Press, 2019) , chapter 59. The reader is supposed to massage equation (3) into equation (4) with the help of the given polylogarithm formulas (1) and (2). I do not see at all how that's possible...
  24. P

    I An integral rewritten (from “Almost impossible integrals“, p.59 in Valean)

    I want to understand where the minus 1 in the first line in the RHS term comes from. I assume the little apostrophe means taking a derivative. But the antiderivative of x^(n-1) is (1/n)x^n. Why the -1? thank you
  25. Prez Cannady

    Looking for a textbook introduction to integrals of the following form

    Summary:: Pretty sure they have something to do with path integrals, or what not. But obviously it's hard to *search* for this stuff. Basically, I'm looking for a textbook, any textbook--physics, mathematics, etc.--that deals with integrals that look something like this (mistakes are mine): S...
  26. Adesh

    Why does dividing by ##\sin^2 x## solve the integral?

    If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...
  27. Math Amateur

    I The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...
  28. Math Amateur

    MHB The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...
  29. CaptainX

    I Surface Integrals: Finding Centroid & Inertia of Circle

    How to find the centroid of circle whose surface-density varies as the nth power of the distance from a point O on the circumference. Also it's moments of inertia about the diameter through O. I'm getting x'=2a(n-2)/(n+2) And about diameter -4(a×a)M{something}
  30. SchroedingersLion

    Why Does My FFT Integration Method Produce Nonzero Imaginary Parts?

    Hi guys, for a project I had to get involved with discrete Fourier transforms to solve PDEs. However, the code that I implemented according to a pseudo-code from a paper did not work - it seems like I calculated integrals incorrectly. To search the error, I tried to integrate the sin(x)...
  31. karush

    MHB Triple Integral: $\dfrac{\sqrt{1-x^2}}{2(1+y)}$

    ok this is a snip from stewards v8 15.6 ex hopefully to do all 3 here $\displaystyle\int_0^1\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}\dfrac{z}{y+1} \,dxdzdy$ so going from the center out but there is no x in the integrand $\displaystyle\int_0^{\sqrt{1 - x^2}} \dfrac{z}{y + 1}dx =\dfrac{ \sqrt{1 -...
  32. karush

    MHB Explaining the Concept of Triple Integrals in Calculus

    15.6.4 Evaluate the iterated integral $$\int_0^1\int_y^{2y}\int_0^{x+y} 6xy\, dy\, dx\, dz$$OK this is an even problem # so no book answer but already ? by the xy
  33. A

    I Overlap Integrals: Understand & Learn from a Source

    Hi I study optics and many times i found a term called (Overlap integral ) as attached pictures .. I can't understand from where these expression comes (mathematically) and what these functions means in particular ( even from mathematical point of view) I can't understand the nature of...
  34. bob012345

    A Coulomb and Exchange Integrals

    Summary: I'm looking for a table of Coulomb and exchange integrals for Lithium and beyond. I'm looking for a convenient table of Coulomb and exchange integrals for Lithium and beyond. I've looked everywhere and I find integrals for J,K 1s, 2s or 1s,2p for Helium. Does anyone know of a...
  35. Beelzedad

    I Swapping the order of surface and volume integrals

  36. Beelzedad

    I Is interchanging the order of the surface and volume integrals valid here?

    Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. Consider the following multiple integral: ##\displaystyle A=\iiint_{V'} \left[ \iint_S \dfrac{\cos(\hat{R},\hat{n})}{R^2} dS \right] \rho'\ dV' =4 \pi\ m_s## where...
  37. karush

    MHB 2.6.62 inverse integrals with substitution

    ok this is from my overleaf doc so too many custorm macros to just paste in code but I think its ok,,, not sure about all details. appreciate comments... I got ? somewhat on b and x and u being used in the right places
  38. Sophrosyne

    B Speed of light with quantum path integrals

    Richard Feynman formulated quantum path integrals to show that a single photon can theoretically travel infinitely many different paths from one point to another. The shortest path, minimizing the Lagrangian, is the one most often traveled. But certainly other paths can be taken. Using single...
  39. Mondayman

    Calculus Where Can I Find More Challenging Integrals for My Competition?

    Hi folks, I love doing integrals, and I think I'm going to start a competition at my school. The integrals in the standard calculus textbooks I have access to, Briggs, Stewart, etc., are pretty elementary. I am looking for some harder integrals. I have the books Irresistible Integrals and...
  40. Hiero

    I Vector valued integrals in the theory of differential forms

    So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector? In particular I’m...
  41. saadhusayn

    Perturbation expansion with path integrals

    I expanded the exponential with the derivative to get: ## Z = \Bigg(1 + \frac{1}{2} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} + \frac{1}{4} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl}...
  42. G

    A How does Lorentz invariance help evaluate tensor integrals?

    We're trying to reduce the tensor integral ##\int {\frac{{{d^4}k}}{{{{\left( {2\pi } \right)}^4}}}} \frac{{{k^\mu }{k^\nu }}}{{{{\left( {{k^2} - {\Delta ^2}} \right)}^n}}}{\rm{ }}## to a scalar integral (where ##{{\Delta ^2}}## is a scalar). We're told that the tensor integral is proportional...
  43. J

    A Can the Huygens-Fresnel principle be translated into path integrals?

    Wave optics, including diffraction, seems to be apt for path integral language. In fact, Feynman's double slit language is purely "diffraction". Also, the PDE for the wave equation results in a solution via Green's function, and the Green function is where "the path integral lives". I have...
  44. S

    Comparing summations with integrals

    1. ##\sum_{n=1}^N \arctan{(n)} \geq N \arctan{(N)}-(1/2)\ln{(1+N^2)} \iff \sum_{n=1}^N \arctan{(n)} \geq N \int_0^N \frac{1}{1+x^2} dx - \int_0^N \frac{x}{1+x^2} dx## Where do I go from here? I've tried understanding this graphically, but to no avail. 2. Maybe this follows from finding an...
  45. SchroedingersLion

    Integrals over chained functions

    Good evening! Going through a bunch of calculations in Ashcroft's and Mermin's Solid State Physics, I have come across either an error on their part or a missunderstanding on my part. Suppose we have a concatenated function, say the fermi function ##f(\epsilon)## that goes from R to R. We know...
  46. FQVBSina

    A Bessel's Integrals with Cosine or Sine?

    Hello all, This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as: $$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \...
  47. fazekasgergely

    Infinite series to calculate integrals

    For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
  48. LCKurtz

    Insights Demystifying Parameterization and Surface Integrals

    Continue reading...
  49. S

    I What's the difference between the Riemann & Darboux Integrals?

    I was reading about this, and they seem the same. Of course, if they were the same, they wouldn't have different names.
  50. J

    Integrals of tensor perturbations (Weinberg, p. 315)

    Hi, I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that $$ \int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2), $$ where ##e_{ij}(\hat{q})## is any...
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