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. Math Amateur

    MHB Properties of Contour Integrals - Palka Lemma 2.1 (vi) .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 4: Complex Integration, Section 2.2 Properties of Contour Integrals ... I need help with some aspects of the proof of Lemma 2.1, part (vi), Section 2.2, Chapter 4 ... Lemma 2.1, Chapter...
  2. D

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  3. M

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  4. M

    Proving the equation for the height of a cylinder

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  5. M

    Question about Finding a Force with line integrals

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  6. W

    Contour Integrals: Working Check

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  7. M

    Question about Vector Fields and Line Integrals

    Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...
  8. Marcin H

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  9. Marcin H

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  10. J

    MHB The Residue Theorem To Evaluate Integrals

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  11. Helmholtzerton

    A Z-coordinate shift when using elliptical integrals?

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  12. W

    Solve Integral Equation: xe-axcos(x)dx from 0 to ∞

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  13. C

    I Analytically continuing 2,3,4pt integrals

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  14. karush

    MHB 15.2.87 Write the following integrals as a single iterated integral.

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  15. J

    Definite integrals and Functionals

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  16. G

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  17. harpazo

    MHB Change of Variables in Multiple Integrals

    Let S S = double integrals S S x^2 dA; where R is the region bounded by the ellipse 9x^2 + 4y^2 = 36. The given transformation is x = 2u, y = 3v I decided to change the given ellipse to a circle centered at the origin. 9x^2 + 4y^2 = 36 I divided across by 36. x^2/4 + y^2/9 = 1 I replaced...
  18. S

    I Peskin book on QFT question -- 2 integrals for D(x−y)

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  19. H

    B Definite integrals with +ve and -ve values

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  20. S

    I Merging Two Threads: Complex Integrals & Branch Cuts

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  21. harpazo

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  22. S

    B Some help understanding integrals and calculus in general

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  23. B

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  24. R

    I Confusion regarding line integrals

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  25. harpazo

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  26. harpazo

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  27. N

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  28. harpazo

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  29. R

    I What other math courses involve multiple integrals?

    Didn't know where to put this question. but just wanted to ask quickly. All i know is that in Calculus 3, you use double, triple or multiple integrals for 3D models. But what other higher level math course uses or involves multiple integrals?
  30. R

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  31. S

    I A, a† etc. within integrals: does it have to be so hard?

    I have been trying to teach myself quantum optics for some time. Up to now, I have often looked at certain types of integrals -- the ones that have operators within them -- without going into too much detail, just trying to get the general purport and moving ahead, only to get mired in some...
  32. M

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  33. harpazo

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  34. Mr Davis 97

    I Evaluating improper integrals

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  35. Mr Davis 97

    I Evaluating improper integrals with singularities

    For two improper integrals, my textbook claims that ##\displaystyle \int_0^3 \frac{dx}{(x-1)^{2/3}} = 3(1+2^{\frac{1}{3}})## and that ##\displaystyle \int_0^8 \frac{dx}{x-2} = \log 3##. However, when I put these through Wolfram Alpha, the former exists but the latter does not, and it says that...
  36. D

    Determine the truth of the following statements

    Homework Statement ##f(x) = \begin{cases} -\frac{1}{1+x^2}, & x \in (-\infty,1) \\ x, & x \in [1,5]\setminus {3} \\ 100, & x=3 \\ \log_{1/2} {(x-5)} , & x \in (5, +\infty) \end{cases}## For a given function determine the truth of the folowing statements and give a brief explanation: a) Function...
  37. Kumaken

    Application of integrals to find moment Inertia

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  38. doktorwho

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    What would tou suggest as the best resource for learning integrals? I need preferably some practical books videos or youtube channels that deal with application and problems rather than theory. Any thoughts? Thanks
  39. Ben Wilson

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    Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf). $$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...
  40. stevendaryl

    Insights Solve Integrals Involving Tangent and Secant with This One Trick - Comments

    stevendaryl submitted a new PF Insights post Solve Integrals Involving Tangent and Secant with This One Weird Trick Continue reading the Original PF Insights Post.
  41. C

    I Tensor integrals in dimensional regularisation

    Consider a d dimensional integral of the form, $$\int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma} \ell^{\mu}}{D}\,\,\,\text{and}\,\,\, \int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma}}{D}$$ where ##D## is a product of several propagators. One can reduce this to a sum of scalar integrals by...
  42. A

    I How come surface integrals are single integrals in my book?

    I am currently reading Young & Freedmans textbook on physics as part of a university course, and I've noticed that they repeatedly represent surface integrals (which are double integrals) as single integrals. For instance, they symbolically represent the magnetic flux through a surface as: \int...
  43. mercenarycor

    Question about Legendre elliptic integrals

    Homework Statement [/B] J(a, b, c;y)=∫aydx/√((x-a)(x-b)(x-c)), let a<b<cHomework Equations f(θ, k)=∫0θdx/√(1-k2sin2(x)), k≤1 The Attempt at a Solution This is an example from my study material, and I don't understand the first step they do. Let x=a+(b-a)t, dx=(b-a)dt Wait...what? Why? How...
  44. MAGNIBORO

    B Small big problem with integrals

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  45. doktorwho

    Integration by substitution question

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  46. cg78ithaca

    A Inverse Laplace transform of a piecewise defined function

    I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...
  47. cg78ithaca

    A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

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  48. D

    A How to Solve the Fourier Integral in Eq. (27) Involving Position Vectors?

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  49. M

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  50. M

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