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.

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

    I Why is this closed line integral zero?

    This problem comes from fluid dynamics where Kelvin circulation theorem states, that if density "rho" is a function of only pressure "p", then closed line integral of grad(p) / rho(p) equals zero. It seems so trivial, so that no one ever gives reason for this claim. When trying to solve it...
  2. PainterGuy

    How do I change this integral limit from x to t?

    Hi, It's not a homework problem. I was just doing it and couldn't find a way to change the integral limit from "x" to "t". I should end up with kinetic energy formula, (1/2)mv^2. I've assumed that what I've done is correct. Thank you! Edit: "E" is work done.
  3. patric44

    I Has the Fermi-Dirac Integral been solved?

    hi guys I have a question about whether or not the Fermi-Dirac Integral has Been solved, because i found a formula on Wikipedia that relates the Fermi-Dirac integral with the polylogarithm function: $$F_{j}(x) = -Li_{j+1}(-e^{x})$$ and in some recent papers they claim that no analytical...
  4. A

    I Index and bound shift in converting a sum into integral

    Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted? $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$
  5. Leo Liu

    What does this integral notation mean?

    I saw it somewhere but I did't know exactly what it meant. Could someone explain it to me like I am 5? Does it mean we integrate with respect to x n times? $$\int_{\mathbb{R}^n}f\, \mathrm{d}^n x$$
  6. W

    Volume of solid region double integral

    I sketched this out. With the z=0 and y=0 boundaries, we are looking at ##z \geq 0## and ##y \geq 0## I believe ##0 \leq x \leq 5## because of the boundary of ##y=\sqrt{25-x^2}##. This is my region ##\int_0^5 \int_0^\sqrt{25-x^2} x \, dydx ## ## =\int_0^5 xy \vert_{0}^{\sqrt{25-x^2}} \, dx##...
  7. patric44

    I Integral representation of incomplete gamma function

    hi guys I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by $$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$ i was thinking about taking substitutions in order to...
  8. RicardoMP

    A How to reduce an integral in phase space to a one-dimensional form?

    I've been trying for a very long time to show that the following integral: $$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2 f_{L}^0f_{R}^0(1+f_{H}^0). $$ can be reduced to one dimension: $$ I_D...
  9. LCSphysicist

    Understanding Scattering Process in QFT Integral

    I have been studying scattering process in QFT, but i am stuck now because i can't understand how this integral was evaluated: $$\int dp\space \frac{1}{\sqrt{p^2+c²}}\frac{1}{\sqrt{p^2+k²}}\space p² \space d\Omega \space \delta(E_{cm}-E_{1}-E_{2})$$$ Where Ecm = c + k, E1 is the factor in the...
  10. K

    I How do I find the expected value and median of a probability density function?

    Hey everyone, I have been struggling to find the expected value and median of f(x) = 1/2e^-x/2, for x greater than 0. I am just wondering how I do so? Thank you.
  11. jk22

    I Why the integral of a differential does not give the function back in 2D?

    Let f be a 2 variables function. 1) ##f(x,y)=g(x)+h(y)\Rightarrow df=g'(x)dx+h'(y)dy\Rightarrow\int df=g(x)+k(y)+h(y)+l(x)=f(x,y),\textrm{ if } k=l=0## 2) ##f(x,y)=xy\Rightarrow df=ydx+xdy\Rightarrow\int df=2xy+k(y)+l(x)\neq f(x,y)## Why in the second case the function cannot be recovered ?
  12. Eclair_de_XII

    Converting integration of rectangular integral to spherical.

    I'm going to type out my LaTeX solution later on. But in the meantime, can anyone check my work? I know it's sloppy, disorganized, and skips far more steps than I care to count, but I'd very much appreciate it. I'm not getting the answer as given in the book. I think I failed this time because I...
  13. LCSphysicist

    Calculus: Integral along a curve.

    Let $F = (P(x,y),Q(x,y))$ a field of vector class 1 in the ring $A={(x,y): 4<x²+y²<9}$ and $x,y$ reals. I am having trouble to understand why this alternative is wrong: If $ \partial P /\partial y = \partial Q /\partial x$ for every x,y inside A, so $\int_{C} Pdx + Qdy = 0$ for every...
  14. MountEvariste

    MHB Definite integral involving sine and hyperbolic sine

    Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$
  15. Flamitique

    I Solving an Integral using Feyman's trick

    Hey guys ! I just need a little help on a integral I was trying to solve using feyman's technique. This is the integral from 0 to 1 of (sin(ln(x))/ln(x) dx, which has been solved in one of the videos of bprp, but I'm trying to solve it using a different technique, and I end up with a different...
  16. ?

    Please evaluate this double integral over rectangular bounds

    Summary:: Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine. [Mentor Note -- thread moved from the technical math forums, so no Homework template is shown] Hi, I'm trying to find the answer to the following integral over the rectangle...
  17. brotherbobby

    The integral of a function ##f(x)## from its graph

    Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##. Attempt : I argue for or against each statement by putting it down first in blue and my answer in red. ##g(x)## is always positive : The exact...
  18. L

    Solving this integral with respect to parameter m

    It is clear that ##1-x^2## is equal to zero in both boundaries ##1## and ##-1##. So for me is interesting to think like this \frac{d^m}{dx^m}(1-x^2)^m=\frac{d}{dx}(1-x^2)\frac{d}{dx}(1-x^2)\frac{d}{dx}(1-x^2)... and...
  19. A

    I What is the indefinite integral of Bessel function of 1 order (first k

    Hi When we find integrals of Bessel function we use recurrence relations. But this requires that we have the variable X raised to some power and multiplied with the function . But how about when we have Bessel function of first order and without multiplication? How should we integrate it ?
  20. A

    How to Fix Limits of Integration for Gamma Integral #6?

    Hi I have a gamma integral in which it is not obvious how I can fix the limits of integration in order to match the standard form of gamma function.I just need someone to tell me how to fix them. I mean the integral number 6 in the picture. You can see my attempt in the PDF .
  21. B

    I Is there a way to simplify this integral involving an exponential function?

    Hello! I have a function ##f(t)## such that ##\int_a^b{f(t)dt}=f_0##. Is there a way to calculate (or bring it to a simpler form) ##\int_a^b{f(a)e^{t}}dt##? Thank you!
  22. greg_rack

    Problem solving a parametric indefinite integral

    Since ##h## and ##k## are constants: $$\frac{h}{k}\cdot \int \frac{1}{y(h-y)} \ dy$$ Now, rewriting the integrating function in terms of coefficients ##A## and ##B##: $$\frac{1}{y(h-y)}=\frac{A}{y}+\frac{B}{h-y}\rightarrow B=A=\frac{1}{h} \rightarrow$$ $$\frac{1}{h}\int \frac{1}{y}\ dy +...
  23. A

    MHB How can I integral this problem?

    Question \[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \] How can I derive this answer?
  24. stevendaryl

    I Integral involving exponential

    Just a quick question: Does anybody know if there is a closed-form solution to this rather simple-looking definite integral? ##F(\lambda) = \int_0^{\infty} \dfrac{e^{-x}}{1 + \lambda x} dx## If ##\lambda > 0##, it definitely converges. It has a limit of 1 as ##\lambda \rightarrow 0##. But it...
  25. LCSphysicist

    How to perform a integral in momentum space

    I am not sure how does the integral was did here. More preciselly, How to go from the first line to the second line? Shouldn't it be $$\frac{4 \pi}{(2 \pi)^3} \int _{0} ^{\infty} p^2 e^{ip*r}/(2 E_p)$$ ? (x-y is purelly spatial)
  26. docnet

    How do I evaluate this integral?

    The goal is to evaluate the below integrals. Please note ##x\in \mathbb{R}^3## The issue is that I do not understand the meaning of the integration boundary ##||y-x||=t## and the meaning of the notation ##dS(y)##. Would someone be kind to explain these notations to me like I am five? are ##x##...
  27. C

    Vector calculus - show that the integral takes the form of (0, a, 0)

    Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right. I also tried to compute the right-hand side of...
  28. Addez123

    Solve p = P(2X <= Y^2) using double integral

    Background information Earlier they've shown that some double integrals can be simulated if it contains pdfs. Ex: $$\int \int cos(xy)e^{-x-y^2} dx dy$$ Can be solved by setting: Exponential distribution $$f(x) = e^{-x}, Exp(1)$$ Normal distribution $$f(y) = e^{-y^2}, N(0, 1/\sqrt 2)$$By knowing...
  29. docnet

    Help computing the following integral

    Solution attempt: we make the substitution ##\frac{s}{2}=u## and ##ds=2du## to compute...
  30. greg_rack

    Apparently impossible indefinite integral?

    Hi guys, I got to solve this integral in a recent test, and literally I had no idea of where to start. I thought about substituting ##tan(\frac{x}{2})=t## in order to apply trigonometry parametric equations, integrating by parts, substituting, but I always found out I was just running in a...
  31. Leo Liu

    Find the bounds after changing the variables in a double integral

    The answer calculates the integral with ##du## before ##dv## as shown below. However I decided to compute it in the opposite order with different bounds. Here is my work: According to the definitions, $$\begin{cases} u=x+y\\ v=2x-3y \end{cases}$$ First we need to convert the boundaries in xy...
  32. JD_PM

    I Understanding how to derive the Feynman rules out of the path integral

    I am studying interacting scalar fields (from Osborn) using the path integral approach. We define the functional integral \begin{equation*} Z[J] := \int d[\phi] e^{iS[\phi] + i\int d^d x J(x) \phi(x)} \tag{1} \end{equation*} The idea is to differentiate ##Z[J]## with respect to ##J## and end...
  33. A

    Does a double principal-value integral exist?

    Encountered this integral and I believe it converges by studying it numerically but not sure and was wondering how might I show it converges or diverges? Surely there must be a way. $$ \text{P.V.}\int_0^{\infty}\int_0^{\infty} \frac{\text{sinc}^2(1+\phi)v e^{-v}}{(e^{v/2}-1)(\phi-v)}dvd\phi $$...
  34. greg_rack

    Solving an immediate indefinite integral of a composite function

    That's my attempt: $$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$ Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets: $$-\frac{1}{2tan^2x}+c$$ But there is something wrong... what?
  35. A

    Problem showing dilogarithm integral is -pi^2/6

    I am working with the Dilogarithm function and am having problems showing the following and was wondering if someone could help: $$ \int_0^1\int_0^y\left(\frac{1}{x-1}\right)\left(\frac{1}{y}\right)dxdy=-\frac{\pi^2}{6} $$ This is what I have so far: Iterating the first level: $$ \begin{align*}...
  36. chwala

    Find the derivative of given function and hence find its integral

    ##y=x^2ln x-x## ##\frac {dy}{dx}=2x ln x+x-1## ##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x## it follows that, ##\int [2x ln x +x]\,dx##=##x^2 ln x## ##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx## ##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##
  37. D

    Volume integral of x^2 + (y-2)^2 +z^2 = 4 where x , y , z > 0

    (a) i sketched a quarter of a sphere centred at x=0 , y=2 , z=0 (b ) ∫ ∫ √ (4-x2 - (y-2)2) dx dy with limits 0 < x < 2 and 0 < y <4 (c ) i converted to spherical polars and took the integrand as 1/r2 . the volume element is r2sinθ drdθd∅ This leads to the triple integral of sinθ with...
  38. Arman777

    Python Solving an Integral equation with uncertainties

    I have some variables that are uncertain, these are w_m = u.ufloat(0.1430, 0.0011) z_rec = u.ufloat(1089.92, 0.25) theta_srec = u.ufloat(0.0104110, 0.0000031) r_srec = u.ufloat(144.43, 0.26) and some constant values c = 299792.458 # speed of light in [km/s] N_eff = 3.046 w_r = 2.469 *...
  39. Arman777

    I Error propagation of a variable for an integral

    I have an integral that depends on two parameters ##a\pm\delta a## and ##b\pm \delta b##. I am doing this integral numerically and no python function can calculate the integral with uncertainties. So I have calculated the integral for each min, max values of a and b. As a result I have obtained...
  40. A

    Can we use the disk method in this integral?

    Goodd day, I have a question regarding an exercice I have already posted Bvu was very nice and provided this darwing I already have the solution But y question is : can we use the disk method? because as you can see even though the intersection was at x=-1 the sphere goes deep into the...
  41. A

    Problem with a triple integral in cylindrical coordinates

    Good day here is the solution J just don't understand why the solution r=√2 has been omitted?? many thanks in advance best regards!
  42. Arman777

    Is there a Python function that finds an unknown inside an integral?

    I have a integral with unknown h. My integral looks like this where C, a, b are constants F(x) and G(x) are two functions. So the only unknows in the integral is h. How can I solve it ? I guess I need to use scipy but I don't know how to implement or use which functions. Thanks
  43. A

    I What does this integral represent?

    I cannot understand what this integral is doing: $$g(x)=\left(\frac{i \pi}{2}-\gamma\right) f(x)+\frac{1}{2}\,\text{P.V.}\int_{-\infty}^\infty \left(\frac{1}{x-x'}-\frac{1}{| x-x'| }\right)\,f(x')\,dx'$$ Can anybody please rewrite it in a more understandable form?
  44. L

    A Integral -- Beta function, Bessel function?

    Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?
  45. P

    Double Integral via Appropriate Change of Variables

    Summary:: Calculate a double integral via appropriate change of variables in R^2 Suppose I have f(x,y)=sqrt(y^12 + 1). I need to integrate y from (x)^(1/11) to 1 and x from 0 to 1. The inner integral is in y and outer in x. How do I calculate integration(f(x,y)dxdy) ? My Approach: I know that...
  46. A

    MHB Multivariate gaussian integral

    I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $\times$ 1 and T is a positive definite symmetric p $\times$ p matrix. The integral is the...
  47. S

    Integral as approximation to summation

    Writing down several terms of the summation and then doing some simplifying, I get: $$\sum_{r=1}^n \frac{1}{n} \left(1+\frac{r}{n} \right)^{-1}= \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...\frac{1}{2n}$$ How to change this into integral form? Thanks
  48. K

    Basel Problem Integral: Solving with Calculus

    Summary:: Using an integral and taylor series to prove the Basel Problem The Basel problem is a famous math problem. It asked, 'What is the sum of 1/n^2 from n=1 to infinity?'. The solution is pi^2/6. Most proofs are somewhat convoluted. I'm attempting to solve it using calculus. I notice...
  49. T

    A Dx in an integral vs. differential forms

    Good Morning To cut the chase, what is the dx in an integral? I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx That said, I have seen it in an integral, specifically for calculating work. I do understand the idea of...
  50. murshid_islam

    I Gaussian integral by differentiating under the integral sign

    Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...
Back
Top