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.

View More On Wikipedia.org
Recent contents Top users
  1. SamRoss

    B Can you help me see why these integrals are the same?

    I am reading "Inside Interesting Integrals" by Paul Nahin. Around pg. 59, he goes through a lengthy explanation of how to do the definite integral from 0 to infinity of ∫1/(x4+1)dx. However, he then simply writes down that this integral is equal to ∫x2/(x4+1)dx with the same limits. Now, it's...
  2. matai

    Using Integrals to Calculate the Rotational Energy of Earth

    So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that...
  3. D

    Python program to calculate Kirkwood−Buff Integrals

    I want to write a python program to calculate the Kirkwood−Buff Integrals. The equation is as: G=4π∫0 ∞ r 2 [g(r)-1] dr I have the g(r) values. Any suggestions are highly appreciated. Thank you.
  4. F

    Mathematica Problem with line integrals in Mathematica

    Hello everyone. I am testing mathematica to work with some line integrals. I want to go from the point (0,0) to (2, 3) over a straight line. I do it with 3 different parametrizations. The problem is that each one offers me a different result. The original problem is a two dimensional gaussian...
  5. YoungPhysicist

    B A rookie question for integrals of polynomial functions

    $$\int x^2+3 = \frac{x^3}{3}+3x+C$$ I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p
  6. Hiero

    How would we integrate this without using double integrals?

    Homework Statement [/B] Evaluate ##\int_0^∞ \frac{\tan^{-1}(\pi x) - \tan^{-1}x}{x}dx ## 2. Relevant information This problem comes after a chapter on “multiple integrals” and so, in this context, I realized we could rewrite the single integral as a double integral: $$ \int_0^∞ \int_1^{\pi}...
  7. benorin

    B Does QM Use n-Dimensional or Infinite Dimensional Integrals?

    Much as the title of this thread asks: does quantum mechanics use n-dimensional or infinite dimensional integrals? I'm merely curious as I studied the n-dimensional case as a hobby and wondered if I'd ever get to use it for anything cool like QM. If so, please maybe post one such integral so I...
  8. mishima

    I Help finding more info on some theorems [Vector Integrals]

    Hi, in Boas Mathematical Methods in the Physical Sciences, Chapter 6 section 11 problem 17 has a list of 7 theorems it calls "Vector Integral Theorems". For example, $$\int \vec \nabla \times \vec V \ d\tau = \oint \vec n \times \vec V \ d\sigma$$ I understand their derivations from the...
  9. W

    Probability Theory: Order statistics and triple integrals

    Homework Statement Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##. a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##. b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...
  10. R

    Integral of Acceleration with respect to time

    Homework Statement Acceleration is defined as the second derivative of position with respect to time: a = d2x/dt2. Integrate this equation with respect to time to show that position can be expressed as x(t) = 0.5at2+v0t+x0, where v0 and x0 are the initial position and velocity (i.e., the...
  11. Jazzyrohan

    I Change of order in double integrals

    In the question given below, can we change the order of integral so that y can be the independent variable and x be the dependent one?The cylinder x^2 + z^2 = 1 is cut by the planes y=0,z=0 and x=y.Find the volume of the region in the first octant.This may look like a homework question but it's...
  12. R

    I Integrals: Math & QM Definitions

    Math definition: integral of function within limits divided by difference of limits. QM definition: integral of complex conjugate of wave equation times function times wave equation within limits of minus to plus infinity.
  13. K

    I need some help with integrals

    Homework Statement You are given the function f(x)=3x^2-4x-8 a) Find the values of a. Explain the answers using the function. Homework EquationsThe Attempt at a Solution a^3-2*a^2-8*a=0 a=-2 v a=0 v a=4 I found the answers, but I don't know how to explain my answers by using the function...
  14. starstruck_

    I Double integrals (line vs. Area)

    Hey! So we were doing double integrals in electricity and magnetism for vectors dA and A (for electric flux). I’m a little confused. Doing a double integral of vectors dx and dy gave an area (vector) dA and A. Thinking back to calc 1, when we had FUNCTIONS (not vectors) they gave the area...
  15. A

    MHB Reference request for self-studying multiple Riemann integrals

    As the title says, I would like to self-study multivariable real analysis (integration, specifically; the Riemann integral) and I need some recommendations (resources, books, videos, ...). I'm from Croatia and got my hands on some Croatian notes about multivariable real analysis so if some of...
  16. M

    Mathematica Decimals give different integrals than fractions; why?

    Just like the title says. Is this due to roundoff?
  17. H

    A How to evaluate path integrals numerically?

    Since we only know Gaussian integration, could one get Green's function numerically with interacting action. Usual perturbation theory is tedious and limited, could one get high accurate result with PC beyond perturbation?
  18. Bill2500

    I Munkres-Analysis on Manifolds: Extended Integrals

    I am studying Analysis on Manifolds by Munkres. He introduces improper/extended integrals over open set the following way: Let A be an open set in R^n; let f : A -> R be a continuous function. If f is non-negative on A, we define the (extended) integral of f over A, as the supremum of all the...
  19. W

    I Integrals are harder than derivatives, why?

    I understand the concept of derivatives but when it comes to integrals and their uses I do not understand what they do and where you use them.In derivatives you can understand how a function changes but in integration everything is so illogical.Can someone explain me the use of integrals in...
  20. W

    Integrating Over Two Variables: Deriving an Expression for dP/dt

    Homework Statement Hi everyone, I'd appreciate it if someone could help look through my working and check if it makes sense! I have the following integral: $$P(t) = \int_{-\infty}^{a} \int_{-\infty}^{-\infty} f_p(p) \ f_{x} (x - pt/m) \ dp \ dx$$ I want to find an expression for...
  21. M

    Dx before the f(x) in integrals

    Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
  22. prakhargupta3301

    Having a problem in steps while solving integrals

    Homework Statement The problem is attached. I'm new to these problems (calculus). I'm not getting my answer as any of the options. I need your help to know whether me or the slide is wrong. Homework Equations x_x[/B]The Attempt at a Solution Thank you for reading.
  23. prakhargupta3301

    Having a problem in steps while solving integrals

    Homework Statement My problem is in integral calculus (I'm new to it). I know what it is and how it works (basically. I'm not too advanced right now). The problem is as following: (I will be posting comments/reasons along with what I've done and with what logic/understanding I've done it...
  24. prakhargupta3301

    Having a problem in steps while solving integrals

    Okay. So I'm new to calculus. And this is the first time I'm solving a physics problem using integration. I understood that ∫dt (or ∫1dt) will be equal to just 't+C.' (Just like f ' (t+C) = 1). Though, that's not the problem. The problem is when I apply it this way: Question: The expression for...
  25. M

    MHB Find Velocity of Particles: Indefinite Integrals

    To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration function, a(t), i.e. v = Z a(t) dt. Your help greatly appreciated.
  26. karush

    MHB 15.4.20 volumn via triple integrals

    $\textsf{The region in the first octant bounded by the coordinate planes and the surface }$ $$z=4-x^2-y$$ $\textit{From the given equation we get}$ \begin{align*}\displaystyle &0 \le z \le 4-x^2-y\\ &0 \le y \le 4-x^2\\ &0 \le x \le z \end{align*}...
  27. T

    A simple case of translation invariance of Riemann integrals

    Homework Statement Show that \int_{A} 1 = \int_{T(A)} 1 given A is an arbitrary region in R^n (not necessarily a rectangle) and T is a translation in R^n. Homework Equations Normally we find Riemann integrals by creating a rectangle R that includes A and set the function to be zero when x...
  28. W

    Problem with line integrals for electric potential

    Homework Statement I have a problem understanding the equation $$\Delta V = -\int_{a}^{b} \vec{E} \cdot d \vec{l}$$ In the case of a parallel plate capacitor whereby the positive plate is placed at ##z=t## while the negative is at ##z = 0##, my integral looks like $$\Delta V = -\int_{0}^{t}...
  29. rocdoc

    I Path Integrals in Quantum Theory

    I have found a general result for certain exponential integrals that may be of interest to those involved with using path integrals. I am not certain that I am applying it correctly but it appears to work, and I can reproduce results quoted in various textbooks , using it. This may however be...
  30. R

    MHB Spherical coordinates and triple integrals

    Suppose $\displaystyle f = e^{(x^2+y^2+z^2)^{3/2}}$. We want to find the integral of $f$ in the region $R = \left\{x \ge 0, y \ge 0, z \ge 0, x^2+y^2+z^2 \le 1\right\}$. Could someone tell me how we quickly determine that $R$ can be written as: $R = \left\{\theta \in [0, \pi/2], \phi \in [0...
  31. W

    Maple Computing Numerical Integrals with Maple

    Hi all, I am new to the Maplesoft software and have been experiencing trouble computing numerical integrals. I defined a few mathematical functions in terms of a few variables like so: I then used "subs" to input values to anything that isn't already a defined constant (like ##\hbar,\pi## and...
  32. D

    Faraday's Law--Confusion about terms in the integrals

    Hi. The induced emf is given by -d/dt ∫B.dS but when the time derivative is taken inside the integral sign this becomes -∫ ∂B /∂t.dS . Why isn't B.dS differentiated using the product rule giving an extra term inside the integral sign ? For some reason the integral sign is appearing as a small...
  33. F

    I Improper Integrals - Are They Really Integrals?

    I understand what improper integrals are, but are they really integrals? The semantics are just a bit confusing.
  34. F

    I Improper Integrals: Definite & Indefinite | Bounds -1 to 1

    if I wanted to take the definite integral of 1/x with respect to x, with the bounds -1 and 1, the integral would be improper. What about the indefinite integral? We can find the indefinite integral of 1/x to be ln|x|. Can we find the indefinite integral of discontinuous functions?
  35. Saurabh

    Solve Hairy Trig Integral: Find Value of 'c

    <Moderator's note: Moved from a technical forum and thus no template.> where a, b, c, d and n, all are positive integers. Find the value of 'c'. ------------------------------- I don't really have a good approach for this one. I just made a substitution u = sinx + cosx I couldn't clear up...
  36. MermaidWonders

    MHB Integrals & Limits: Intuitive Understanding of Convergence

    Suppose that $\int_{-\infty}^{\infty} f(x)\,dx$ converges. Then $\lim_{{x}\to{-\infty}}f(x) = \lim_{{x}\to{\infty}}f(x)$. Why is it true? I have some trouble understanding this intuitively.
  37. karush

    MHB Evaluate the spherical coordinate integrals

    $\textsf{Evaluate the spherical coordinate integrals}$ \begin{align*}\displaystyle DV_{22}&=\int_{0}^{2\pi}\int_{0}^{\pi/4}\int_{0}^{2} \, (\rho \cos \phi) \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \\ %&=\color{red}{abc} \end{align*} so then next ...
  38. ertagon2

    MHB Integration by parts, Partial fraction expansion, Improper Integrals

    - check if right check if right Now, 2 seems to be the right answer for A yet when i made x=5 and subtracted new form form the old one I got a difference of ~$\frac{4}{9}$ (should be 0 obviously) I got A=2 B=$\frac{45}{21}$ C=2 How to calculate $\lim_{{x}\to{\infty}}(- e^{-x})$
  39. Eclair_de_XII

    Calculus I need some practice problems involving double integrals

    I'm in my Probabilistic Models in Biology class, and the professor just brought up the subject of double integrals. I haven't done Calculus IV since fall of 2016, so naturally, I kind of forgot how to do integrals with more than one integrand. Can anyone recommend me any (hopefully, free) pdf's...
  40. C

    Calculus 2 - Trig Integrals Question (Integrating cos^2x)

    1. Here's the problem on trig integrating that I'm struggling with (Calculus 2 btw) 2. Wanted to see if I did everything right so far and what to do after all this. The part where I'm stuck is how to integrate (integral)cos^(2)udu and (integral)cos^(2)usin^(2)udu. I'm sure these are easy...
  41. RJLiberator

    Calculating Double Integrals over Two-Dimensional Sets

    Homework Statement For every two-dimensional set C contained in R^2 for which the integral exists, let ##Q(C) = \int \int_C (x^2+y^2) dxdy##. If ## C_1 = [{(x,y) : -1 ≤ x = y ≤ 1}], ## find Q(c).Homework EquationsThe Attempt at a Solution This was a tougher one for me (the other 2 on this...
  42. GaussianSurface

    Calculating distance from speed

    Homework Statement The speed of a runner increased during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. It follows the image's square. Homework Equations...
  43. W

    Writing integrals in terms of the error function

    Homework Statement I have the following integral, $$\frac{1}{\sigma \sqrt{2\pi} t} \int_{-\infty}^{0} \exp[\frac{-1}{2\sigma ^2} (\frac{x-x_0}{t} - p_0)^2]dx$$ that I wish to write in terms of the error function, $$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-g^2}dg$$ However, I can't seem...
  44. M

    MHB Calculate directly the curve integrals

    Hey! :o We consider the space $D$ that we get if we remove from the square $[-2,7]\times [-3,6]$ the open discs with center the point $(0,0)$ and radius $1$ and with center $(3,3)$ and radius $2$. I want to calculate $$\sum_{j=1}^3\oint_{\sigma_j}\left...
  45. S

    I Help with simplifying series of hyperbolic integrals

    Hello. I have this function ## v(x) = -\sum_{i=1} x^i \sqrt{2}^{i-2} \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm## which I can not seem to figure out how to simplify.I tried looking at some partial integration but repeated integration of ## \cosh ## gives polylogarithms which seemed to...
  46. D

    MHB Short table of integrals (.tex file) will include pdf of output.

    \documentclass[12pt]{article} \usepackage{graphicx} \addtolength{\textwidth}{1.5in} \addtolength{\hoffset}{-1.in} \addtolength{\textheight}{2.0in} %\addtolength{\voffset}{-1.68in} \addtolength{\voffset}{-0.8in} \newcommand {\rreal}{\mbox{$R\!\!\!\!\!l\,\,\,\,$}}...
  47. O

    A Question about QFT Diagrams and their Integrals

    I am studying the terms in the dual Taylor expansion of Z_{1}(J) in \phi^{3} theory, and being introduced to Feynman diagrams in the process. I thought I would try to simplify one of the terms in the expansion so that, after taking derivatives of all the sources, I ended up with integrals that...
  48. B

    Fock states as integrals of coherent states

    Edit: I'm pretty sure I have answered my own question. I think I need to sandwich the integral between a bra and ket to pick out one term from the sum. 1. Homework Statement Show that a Fock state ##|n\rangle## can be represented by the integral $$|n\rangle = \frac{\sqrt{n!}}{2 \pi r^n}...
  49. Math Amateur

    MHB Contour Integrals - Example 2.5, Palka, Section 2.2, Ch.4 .... ....

    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 an aspect of Example 2.5,Section 2.2, Chapter 4 ... Example 2.5, Chapter 4 reads as follows:In...
  50. Math Amateur

    MHB Proof of Lemma 2.1, Part (vi) in Palka's Ch.4: Explaining Inequality

    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 some further help with some other aspects of the proof of Lemma 2.1, part (vi), Section 2.2, Chapter 4 ...
Back
Top