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. D

    I Definite integral is undefined and not undefined

    Hi If i calculate the definite integral between the limits of L and 0 of sin(nπx/L)sin(kπx/L) using the trig formula 2sinAsinB = cos (A-B) - cos (A+B) it is undefined when n=k because (n-k) appears in the denominator. If i calculate the same integral with n=k using the formula sin2(nπx/L) = (...
  2. H

    I Hypergeometric Integration

    I'm trying to calculate the volume of a truncated hypersphere. As part of it I want this integral. Clearly when x=1 the integrand is zero. But plugging this into the series give me a number greater than one. It is true that the series is not defined for x=1, but subtracting some tiny sum...
  3. S

    I Area under the curve of a temperature-time graph -> energy?

    Hello everyone, hope you are all well. I have the following problem: I have a temperatur-time graph. If you determine the integral of this graph, you get the unit [kelvin*second]. This unit is as far as I know meaningless. Is it possible to mathematically "transform" the area under the curve...
  4. T

    I Integrating a product of exponential and trigonometric functions

    I am looking for a closed form solution to an integral of the form: $$ \int_0^\infty \frac{e^{-Du^2t}u \sin{ux}}{u^2+h^2} du $$ D, t, and h are positive and x is unrestricted. I have tried everything, integration by parts, substitution, even complex integration with residue analysis. I've...
  5. PhysicsRock

    Dipole moment of given charge distribution

    I have come up with a solution, however, I'm not sure whether I'm correct. A fellow student of mine has a different result. I'm gonna show my solution, and hopefully one of you can confirm my result or tell me what I did wrong. $$ \begin{align} p_z &= \int d^3x z \rho(\vec{x}) \notag \\ &=...
  6. D

    Prove that the following integral vanishes

    We use the invariance of the measure under ##p\rightarrow -p## to get $$-\int d^3p\xi^{rT}\mathbf{p}\mathbf{\sigma}\xi^s(a^{r\dagger}_{-p}a^s_{-p}+a^{s\dagger}_{-p}a^r_{-p}) = -\int d^3p\xi^{rT}\mathbf{p}\mathbf{\sigma}\xi^sA(-p).$$ If this pesky ##A(-p)## can be shown to be equal to ##A(p)## or...
  7. Hamiltonian

    I Finding the pdf of a transformed univariate random variable

    The above theorem is trying to find the pdf of a transformed random variable, it attempts to do so by "first principles", starting by using the definition of cdf, I don't understand why they have a ##f_X(x)## in the integral wouldn't ##\int_{\{x:r(x)<y\}}r(X) dx## be the correct integral for the...
  8. H

    Introducing integral in textbooks

    I was very surprised to read the following in Needham, Visual Complex Analysis: "It is therefore doubly puzzling that the Trapezoidal formula is taught in every introductory calculus course, while it appears that the midpoint Riemann sum RM is seldom even mentioned." I was surprised because I...
  9. Rhdjfgjgj

    Find the integral of ∫1/(1+tanx)dx

    I have done one by assuming tanx as u in substitution
  10. mcastillo356

    B Integration by parts of inverse sine, a solved exercise, some doubts...

    Hi, PF, here goes an easy integral, meant to be an example of integration by parts. Use integration by parts to evaluate ##\int \sin^{-1}x \, dx## Let ##U=\sin^{-1}x,\quad{dV=dx}## Then ##dU=dx/\sqrt{1-x^2},\quad{V=x}## ##=x\sin^{-1}x-\int \frac{x}{\sqrt{1-x^2} \, dx}## Let ##u=1-x^2##...
  11. cianfa72

    I Integral curves of (timelike) smooth vector field

    Hi, suppose you have a non-zero smooth vector field ##X## defined on a manifold (i.e. it does not vanish at any point on it). Can its integral curves cross at any point ? Thanks. Edit: I was thinking about the sphere where any smooth vector field must have at least one pole (i.e. at least a...
  12. Euge

    POTW Integration Over a Line in the Complex Plane

    For ##c > 0## and ##0 \le x \le 1##, find the complex integral $$\int_{c - \infty i}^{c + \infty i} \frac{x^s}{s}\, ds$$
  13. Euge

    POTW Estimate of a Principal Value Integral

    For ##x\in \mathbb{R}##, let $$A(x) = \frac{1}{2\pi}\, P.V. \int_{-\infty}^\infty e^{i(xy + \frac{y^3}{3})}\, dy$$ Show that the integral defining ##A(x)## exists and ##|A(x)| \le M(1 + |x|)^{-1/4}## for some numerical constant ##M##.
  14. mcastillo356

    B I need to check if I am right solving this integral

    Hi, PF 1-The elementary integral is ##\displaystyle\int{\displaystyle\frac{1}{a^2+x^2}dx}=\displaystyle\frac{1}{a}\tan^{-1}\displaystyle\frac{x}{a}+C## 2-The example is...
  15. L

    Help using Green’s functions in solving Differential Equations please

    Hi, unfortunately I have several problems with the following task: I have problems with the tasks a, d and e Unfortunately, the Green function and solving differential equations with the Green function is completely new to me In task b, I got the following for ##f_h(t)=e^{-at}##.Task a...
  16. chwala

    Evaluate the given integral

    Method 1, Pretty straightforward, $$\int_{-1}^0 |4t+2| dt$$ Let ##u=4t+2## ##du=4 dt## on substitution, $$\frac{1}{4}\int_{-2}^2 |u| du=\frac{1}{4}\int_{-2}^0 (-u) du+\frac{1}{4}\int_{0}^2 u du=\frac{1}{4}[2+2]=1$$ Now on method 2, $$\int_{-1}^0 |4t+2| dt=\int_{-1}^{-0.5} |4t+2|...
  17. baby_1

    A Obtaining a variable value from a 5-th degree polynomial in the tangent form

    Hello, Please see this part of the article. I need to obtain the ##\rho (\phi)## value after obtaining the c0 to c5 constants of the ##\sigma (\phi)##. But as you can see after finding the coefficients, solving Eq.(1) could be a demanding job(I wasn't able to calculate the integral of Eq(1)...
  18. PeaceMartian

    How to find integrals of parent functions without any horizontal/vertical shift?

    TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift? Say you were given the equation : How would you find : with a calculator that can only add, subtract, multiply, divide Is there a general formula?
  19. casparov

    Help Solve for the normalization constant of this QM integral

    I'm given the wavefunction and I need to find the normalization constant A. I believe that means to solve the integral The question does give some standard results for the Gaussian function, also multiplied by x to some different powers in the integrand, but I can't seem to get it into...
  20. S

    Solving this definite integral using integration by parts

    Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks
  21. George Wu

    A Relativistically invariant 2-body phase space integral

    I encounter a function that I don‘t know in the calculation of Relativistically invariant 2-body phase space integral: in this equation, ##s##is the square of total energy of the system in the center-of-mass frame(I think) I don't know what the function ##\lambda^{\frac{1}{2}}## is. There are...
  22. Euge

    POTW An Integral with Fractional Part

    Evaluate the integral $$\int_0^1 x\left\{\frac{1}{x}\right\}\, dx$$ where ##\{\frac{1}{x}\}## denotes the fractional part of ##1/x##.
  23. I

    Integrate disk mass density

    I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function: ρ(r)=ρ0*e^(-r/h) A double integral in polar coordinates should do, but im not sure about the solution I get.
  24. G

    Computing path integral with real and Grassmann variables

    The first step seems easy: computation of the $\theta$ and $\overline{\theta}$ integrals give $$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \det(\partial_j w_i(x)) \exp{\left(-\frac{1}{2}w_i(x)w_i(x)\right)}.$$ From here, I tried using that $$\det(\partial_j w_i (x)) = \det\left(\partial_j w_i...
  25. N

    A Double integral with infinite limits

    I have the following problem and am almost sure of the answer but can't quite prove it: ##f(y)## is nonnegative, and I know that ##\int_0^{\infty } f(y) \, dy## is finite. I now need to calculate (or simplify) the double integral: $$\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \...
  26. chwala

    Find the value of the definite integral

    Looking at integration today...i will go slow as i also try finish other errands anyway; i am thinking along these lines; $$\int \sqrt{(ax^2+bx+c)} dx=\sqrt{a}\int \sqrt{\left[x+\frac{b}{2a}\right]^2+\left[\frac{4ac-b^2}{4a^2}\right]} dx$$ ... Therefore, $$\int_0^2 \sqrt{(8t^2+16t+16)}...
  27. ergospherical

    I Solving the Difficult Integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##

    Anyone have some ideas to approach the integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##?
  28. YAYA12345

    I Integral Bee Preparation -- Trouble with this beautiful integral

    While I was preparing for an integrals contest, I had a doubt about the following integral, I tried several substitutions but nothing worked.I would appreciate your support for this beautiful integral. $$ \int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...) \ \mathrm{d}x$$
  29. Mayhem

    Contour integral of z⁷

    First I parameterize ##z## by ##z(t) = 5i + (3 + i - 5i)t## such that ##z(0) = 5i## and ##z(1) = 3 + i##, which means that ##0 \leq t \leq 0## traces the entire line on the complex plane. By distributing ##t##, we achieve a parameterized expression of the form ##z(t) = x(t) + iy(t)## $$z(t) = 3t...
  30. crememars

    Finding a definite integral from the Riemann sum

    Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...
  31. T

    A Need help with an integral -- How to integrate velocity squared?

    The integral is this one: ##\int (\dot x)^2 \, dt,## With ##x=x(t). ## I don't know how to solve that integral and I haven't find nothing to read about on how to proceed with this kind of (implicit function?) integrals without having the initial function.
  32. T

    A Non solvable integral? (dx/dt)^2 dt

    The integral is (dx/dt)^2 dt, where x=x(t) so it can't be just x + C. The non linear system for whom wants to know how did I get to that point is: d(dx/dt)/dt = sqrt(a^2+b^2)*sin(x+alfa+phi) - Kd*(dx/dt); where alfa = atan(a/b), phi = constant angle, Kd = constant coefficient. After...
  33. R

    Expressing Feynman Green's function as a 4-momentum integral

    I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
  34. Viona

    Integral with different variables

    I want to do this integral in the picture: where r1 and a are constants. I know I can integrate each part separately. There will be an integral with respect to r2 multiplied by integral with respect to theta2 and the last one with respect to phi2. But the term under square root confuses me. Can...
  35. Leo Liu

    How to take the double integral of a data set with respect to time

    Question: Suppose I have a data file for the acceleration of an object after every ## \Delta t_i##, how do I obtain the displacement of it? Context: Integral in a PID loop, although not exactly what I am asking as one is sum of error: $$\int_0^T \int_0^T \ddot {\vec \theta(t)}dtdt$$ the other...
  36. mcastillo356

    I Express the limit as a definite integral

    Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt: (i) General Riemann Sums Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
  37. chwala

    Integral of e^cosx: Answers Sought

    I just came across this and it seems we do not have a definite answer...there are those who have attempted using integration by parts; see link below...i am aware that ##\cos x## has no closed form...same applies to the exponential function...
  38. ergospherical

    I Solving Stat Mech Integral with Wolfram Alpha

    Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.
  39. Euge

    POTW Definite Integral of a Rational Function

    Evaluate the definite integral $$\int_0^\infty \frac{x^2 + 1}{x^4 + 1}\, dx$$
  40. E

    I Parameter Integration of Bubble Integral

    Referring to this link : https://qcdloop.fnal.gov/bubg.pdf Using Mathematica Integrate command to solve it does not give the result stated here but I am unclear as to how they got to the result in the 4th line. It is clear that the integrand (1st line) can diverge for certain values of the...
  41. M

    A Can you help to solve this integral? (resin viscosity research)

    I have tried WolfarmAlpha but it could help me. Please note this is not a homework exercise. I am a researcher and I am looking to model viscosity development of resin. there I came across with this express :) $$\int{\frac{1}{a\cdot e^{bx}+c\cdot e^{kx}}dx}$$
  42. H

    I Original definition of Riemann Integral and Darboux Sums

    Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...
  43. S

    Integral of x^n using Reimann sums

    We don't need to worry about the n = -1 so we can assume that the function is continuous on any interval [a,b] where a, b are real numbers if I separate my interval into N partitions, then the right side values in my interval are a + \frac{b-a}{N}, a + 2 \frac{b-a}{N}, ... , a + k...
  44. N

    I Triple equation for integral on a graph

    Hi, so I'm trying to find the volume of a shape using integral, I found the equation of one plane in 3D space but the second one is something like that, which I cannot write in integral as a function: ##\frac{2(2x-a)}{a}=-\frac{2(6y-a\sqrt3)}{a\sqrt3}=\frac{2z-a\sqrt3}{a\sqrt3}## In the 3D...
  45. G

    Can a human calculate this without a calculator?

    my notebook says that we can rewrite the integral $$\int {75\sin^3⁡(x) \cos^2⁡(x)dx}$$ as $$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$ however, i have literally no idea how it got to this point, and i unfortunately can't really provide an "attempt at a solution" for this...
  46. Pipsqueakalchemist

    Engineering (material science) Fatigue life prediction using integral

    So for this question, I understand the math but just wanted to be clear on a few things. So I had this question on my midterm but instead of tensile and compressive stresses, it was tensile and tensile stress. I initially thought that the delta sigma in the integral was the maximum stress so in...
  47. Addez123

    Calculate surface integral on sphere

    I'm supposed to do the surface integral on A by using spherical coordinates. $$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$ $$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$ Now I'm trying to do $$\iint A dS = (rsin\theta cos\phi, rsin\theta...
  48. P

    Integral Calculation: Compute l/sqrt(x2+l2)

    I need compute the integral $$(2\pi)^{-3} \int d^3p e^{-l|p|}e^{i \vec{x} \cdot \vec{p}}$$ The problem does not specified the limits of integration The result is $$\frac{1}{\pi^2} \frac{l}{\sqrt{\vec{x}^2+l^2}}$$I saw the references about t-Student and I had not achieved it. I have tried to...
  49. C

    No Limits of Integration for Electric Field Integral?

    For this problem, The solution is, However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q. However, how would you solve this problem with limits of integration? Many thanks!