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. DiracPool

    B Derivative and integral of the exponential e^t

    If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or...
  2. Eclair_de_XII

    I Learn the Truth about Integrals of 1/x: Debunking Common Misconceptions

    Let's take an integral##−\int_1^e\frac{dx}{x}##. On one hand, this is equal to ##-\ln(x)|_1^e##. But on the other, ##−\int_1^e\frac{dx}{x}=\int_1^e\frac{dx}{-x}##. If I assume that the integral of this is ##\ln⁡(-x)|_1^e##, then I'd be really stupid since ##\ln## is not even defined over the...
  3. N

    I Testing for Divergence using the Integral Test

    Hello all, I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the...
  4. karush

    MHB 7.4.32 Evaluate the integral by completing the square

    7.4.32 Evaluate the integral $\displaystyle \int_0^1\dfrac{x}{x^2+4x+13}\, dx$ ok side work to complete the square $x^2+4x=-13$ add 4 to both sides $x^2+4x+4=-13+4$ simplify $(x+2)^2+9=0$ ok now whatW|A returned ≈0.03111
  5. Beelzedad

    I Multiple integral Jacobian confusion

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  6. M

    Ιntegral calculation : (sin(x))^4 * (cos(x))^6

    Summary: Ιntegral calculation : (sin(x))^4 * (cos(x))^6 Hi all, I tried to solve it, but I got stuck. An advice from my professor is to set: x=arctan(t) Τhanks.
  7. S

    Is there a faster way to integrate this fraction?

    Find integration of: \frac {1}{(x+1)(x^2 + x -1)^\frac{1}{2}} What I did: 1. Use completing square method for the term inside the square root 2. Use trigonometry substitution (I use secan) 3. After simplifying, use another trigonometry substitution (I use weierstrass substitution) 4. Use...
  8. N

    A Another complicated integral

    How to solve $$\int_{-\infty}^{\infty} \frac{e^{-iax}coth[sinh[bx]]}{sinh[bx]} dx$$ mathematica gives the result ::idiv: "Integral of E^(-Iax)\ Coth[Sinh[bx]]\ Csch[bx] does not converge on {-\[Infinity],\[Infinity]}." thanks!
  9. N

    A Is Wolfram's Answer to the Integral Problem Wrong?

    I had been trying to aplly the sokhotski–plemelj theorem but with no success. Moreover i replaced exponential function with taylor expansion but i still can not solve the integral. thanks
  10. JorgeM

    I Is this Dirac delta function integral correct?

    I have to integrate this expression so I started to solve the delta part from the fact that when n=0 it results equals to 1. And the graph is continuous in segments I thought as the sumation of integers $$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx $$ From the fact that actually $$ δ(sin(x))=...
  11. J

    I Calculating the expected value of the square of an integral of Brownian Motion

    For a standard one-dimensional Brownian motion W(t), calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$I can't figure out how the middle term simplifies. $$ \mathsf E\left(\int_0^T W_t\mathrm dt\right)^2 = \mathsf E\left[T^2W_T^2\right] - 2T\mathsf E\left[W_T\int_0^T...
  12. Saracen Rue

    I Integral resulting from the product of two functions/derivative functi

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  13. E

    Can't work out integral in polar coordinates

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  14. Physics lover

    Challenging Integral Homework: Attempting x=tanA/b Substitution

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  15. Delta2

    Do Maxwell's equations in integral form imply action at a distance?

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  16. G

    Prove Inequality for Non-increasing Function g(x): Harald Cramer Ex. 4 pg 256

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  17. Beelzedad

    I Is my interpretation of this three dimensional improper integral correct?

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  18. S

    A Does Feynman's path integral ignore alternative histories?

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  19. karush

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  20. C

    MHB Group Ring Integral dihedral group with order 6

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  21. lfdahl

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  22. E

    B Convention when changing integral limits

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  23. Robin04

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  24. PainterGuy

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  25. lfdahl

    MHB Limit of integral challenge of (e^(-x)cosx)/(1/n+nx^2)

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  26. C

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  27. tensaiyan

    Integral of greatest integer function and its graph

  28. dRic2

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  29. M

    I What if the Jacobian doesn't exist at finite points in domain of integral?

    Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...
  30. amjad-sh

    Python Calculating Spin-Loss of a Particle Using Integral Form

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  31. WMDhamnekar

    MHB Computing Triple Integral in 'R'

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  32. A

    Converting a Cartesian Integral to a Polar Integral

    the graph of x= √4-y^2 is a semicircle or radius 2 encompassing the right half of the xy plane (containing points (0,2); (2,0); (0-2)) the graph of x=y is a straight line of slope 1 The intersection of these two graphs is (√2,√2) y ranges from √2 to 2. Therefore, the area over which we...
  33. aligator11

    Multivariable Triple Integral - Calculus Physics/Math Problem

    Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...
  34. Milsomonk

    Did I correctly set up the two loop scalar integral?

  35. E

    I About the Feynman Path Integral and Principle of Least Time

    I don't entirely get why we usually say that only the shortest path contributes in the path integral. If you calculate the volume of nth fresnel zones which is the locus where the path length is between n-1 and n wavelengths from the shortest path in 3 dimensions, they are the same I believe. So...
  36. A

    Is this surface integral correct?

    Problem Statement: Requesting for re check Relevant Equations: Requesting for re check In this eq.A4 putting ##v=Hr+u## the first integrand in eq.A5 is coming as ##H(r(\nabla•u)-(r•\nabla)u+2u)\ne\nabla×(r×u)## Am I right?? Can I request anyone to please recheck it... using this the author...
  37. bottle_shadow

    I Line integral for work done by gravity

    Dear Physics Forums people, My problem lies in understanding how the following line integral, which represents work done by the gravitational force, was calculated Specifically, in the integral after the 2nd = sign, they implicitly used \hat{r}\cdot d\vec{s} = dr I wish to understand what...
  38. M

    Electrostatics: Understanding this "Work Done" Line Integral Question

    I have a quick question about the work done concept here, especially the line integral part of it. So I understand the fact that the work done from getting from point A to B is: \int_{a}^{b} \vec F \cdot d\vec r . However, within the context of electric fields, when we define electrostatic...
  39. amjad-sh

    Numerical method to solve an integral that contains a singularity?

    In fact I'm working on a condensed matter physics paper, where I stumbled with an integral that I need to visualize. The function, Ls I need to visualize is equal to: $$Ls=4\nu^4 \dfrac{\int_{-1}^{1} \dfrac{( 1-u^2)}{(u+\sqrt{u^2-\nu^2})^3} \, du}{\int_{-1}^{1}-u \Big...
  40. Beelzedad

    I Is Leibniz integral rule allowed in this potential improper integral?

    Electric potential at a point inside the charge distribution is: ##\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'## where: ##\delta## is a small volume around point ##\mathbf{r}=\mathbf{r'}## ##\mathbf{r}##...
  41. Robin04

    Calculating a complex integral

    As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?
  42. S

    MHB Improper integral of an even function

    Hi colleagues This is a very very simple question I can show when $f$ is integrable and is even i.e. $f(-x)=f(x)$ then $\int_{-a}^{a} \,f(x)\,dx=2\int_{0}^{a} \,f(x)\,dx$ what about improper integrals of even functions, like the function ${x}^{2}\ln\left| x...
  43. J

    Gauss's law -- Integral form problem

    Problem Statement: The effective charge density of the electron cloud in a hydrogen atom in its quantum mechanical ground state turns out to be given by pnot(e^-(r/rnot)), where pnot is a negative constant (the clouds charge density at r=0) and rnot is a constant (rnot=0.025nm). Use gauss's law...
  44. SamRoss

    I Why wasn't this symbol "swapped"?

    In a certain derivation, the author begins with $${g(-t)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(\omega)}e^{-i\omega t}d\omega$$ and then says he will replace ##t## with ##\omega## and ##\omega## with ##t##. He then writes $${g(-\omega)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(t)}e^{-it\omega...
  45. M

    MHB How to evaluate a double integral over a bounded region?

    how do I evaluate the double integral 2x + y dA over the region R bounded by y = 3x, 2X + y = 5, x = 0, and y = 0
  46. Adgorn

    Solution of a simple integral equation

    I did the first part, it is part (b) that I'm having trouble understanding. For any ##x \lt b##, ##f(x)=0## and ##\int_0^x {f(t)} \, dt = 0## (since ##f## is 0 everywhere from 0 to ##b##), which turns the equation ##\int_0^x f(t) \, dt = (f(x))^2+C## into ##0=0+C##, which implies ##C=0##. But...
  47. Mr Davis 97

    I Why do I get two different values for an integral?

    Suppose ##t \ge 0##. Let ##\displaystyle I(t) = \int_{-\infty}^{\infty}\frac{x \sin (tx)}{x^2+1}~\text{dx}##. Call this form 1. Note that we can also write the integral as $$ \begin{align*} I(t) &= \int_{-\infty}^{\infty}\frac{x \sin (tx)}{x^2+1}~\text{dx} \\ &=...
  48. A

    How to solve a surface double integral?

    Hi I´d like a suggestion about a surface double integral. If I have a sphere x^2+y^2+z^2=4 is on the top of a cardioid r=1-cosθ. The problem is when I solve the integral I got a inverse sine when the answer is a natural logarithm (ln)
  49. M

    Integral question on a polynomial

    At first I was thinking about using the dirac delta function ##\delta(x-1)##, but then I recalled ##\delta \notin L_2[0,1]##. Any ideas? I'm thinking no such function exists.
  50. Arman777

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