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. fazekasgergely

    Infinite series to calculate integrals

    For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
  2. Aleoa

    Deep meaning of the work integral formula

    I want to understand very deeply the meaning of the work integral formula: \int m\frac{d\bar{v}}{dt}d\bar{l} It is not enough for me to know that it was defined in this way, I want to know why it was defined in this way. To start, what is the physical meaning of m\frac{d\bar{v}}{dt}d\bar{l}...
  3. SamRoss

    B Justification for cancelling dx in an integral

    In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...
  4. M

    Integral calculation using areas

    Evaluate the integral using the properties of definite integral and interpreting integrals as areas. ##\int_{-1}^2 (1-2x)dx## I need to see there are two areas and these are the same but one is under x-axis the other is above x-axis so the value of the integral is zero. To see this is...
  5. J

    MHB Integral Over Unit Sphere of Inner Product

    Problem: Prove that for any $x \in R^n$ and any $0<p<\infty$ $\int_{S^{n-1}} \rvert \xi \cdot x \rvert^p d\sigma(\xi) = \rvert x \rvert^p \int_{S^{n-1}} \rvert \xi_1 \rvert^p d\sigma(\xi)$, where $\xi \cdot x = \xi_1 x_1 + ... + \xi_n x_n$ is the inner product in $R^n$. Some thinking... I...
  6. M

    What are the units of the argument "x" for this cos(x) function integral?

    Show that the value of ##\int_0^1\sqrt(1-cosx)dx## is less than or equal to ##\sqrt2## ##1\ge cos x\ge-1## The problem is a worked one but I am just confused by a simple thing. We integrate the function f ##\int_0^1\sqrt(1-cosx)dx in the interval [0,1] but I don't understand that what stands...
  7. M

    Integrating with Double Limits: X or Y? Calculating the Correct Integral

    So i drew sketch. And I do not understand, how to write integral for calculation, which I should use, X or Y on limit? Is one of them right? First answer gives me 65,7 Second 383,4
  8. I

    Physical interpretation of this integral equation involving distance and time?

    I am able to solve the problem however if x was position and t was time how is this problem interpreted? I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.
  9. V

    MHB Solving Integral Confusion: Limits & Periods

    Hi! $$\lim_{n \to \infty }\int_{0}^{n}\frac{dx}{1+n^{2}\cos^{2}x }$$ I found to solution on the internet but I didn't understood it 100%. First, it says that the function under integral has period $pi$.Why pi ? I know that cos function has period $2kpi$ Consequence: $\int_{0}^{k\pi...
  10. G

    Double integral - What are the upper and lower bounds?

    Cant get my head around the order of the upper and lower bounds for this, Is it always the higher take away the lower?
  11. M

    Expressing a limit as a definite integral

    Express the limit ##lim_{n\rightarrow\infty} \sum_{i=1}^n \frac2n\ (1+\frac {2i-1}{n})^\frac13## This is worked example but I would like to ask about the points I don't understand in the book. "We want to intepret the sum as a Riemann sum for ##f(x)=(1+x)^3## The factor ##\frac2n## suggests...
  12. TheMercury79

    Why does Mathematica return twice this value?

    The integral is$$\int_0^4dz\iint xyz~dxdy$$Constricted to the quarter circular disk ##x^2+y^2=4## in the first quadrant. First I switched to polar coordinates and integrated the double integral by first writing it as:$$\int_0^4z~dz \int_0^\frac{\pi}2\int_0^2...
  13. A

    I Integral equation for large surfaces

    We often neglect the terms of a surface integral ##\int_v(\nabla•A)dv=\int_s(A•ds)## for ##s## to be very large or ##v## to be very large, What is actually the reason behind this to neglect??
  14. C

    Value of x that maximises the integral in a numerical integrator

    Consider an integral of form $$\int_a^b dx f(x) g(x).$$ Is it possible to tell a numerical integrator to spit out the value of ##x \in [a,b]## that maximises the value of ##f(x)g(x)##? I'm mostly interested in incorporating this into some code I have for adaptive integrator gsl_qags in C++...
  15. M

    I Geometric meaning of reversing the limits of an integral

    With respect to operations, I understand why an integral is multiplied by -1 when its limits reversed. But integral is geometrically an area so reversing the limits would not be able to change neither how large is the area nor the shape of the area. Would you please explain changing the limits...
  16. M

    I Why do we ignore the contribution to a surface integral from the point r=0?

    Let ##V'## be the volume of dipole distribution and ##S'## be the boundary. The potential of a dipole distribution at a point ##P## is: ##\displaystyle\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{S'}\dfrac{\vec{M'}.\hat{n}}{r}dS'## If ##P\in V'## and ##P\in S'##, the...
  17. JD_PM

    What are the limits of integration for this surface integral?

    I want to compute: $$\oint_{c} F \cdot dr$$ I have done the following: $$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$ I don't know what limits the surface integral will have. Actually, I am not sure what's the surface. May you shed some light...
  18. Saracen Rue

    B Definite Integral Verification

    Hi everyone, I am trying to find the definite integral of a function (see attached image) from 0 to infinity using Wolfram alpha. I'm just looking for some verification on if the integral actual is equal to exactly 1, or if there's some rounding errors going on. Thank you for your time :)
  19. A

    What is the Correct Approach to Solve This Complex Integral?

    I split this to get \begin{equation} \int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx \end{equation} Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...
  20. W

    A Path Integral Approach To Derive The KG Propagator

    I'm having trouble understanding a specific line in my lecturers notes about the path integral approach to deriving the Klein Gordon propagator. I've attached the notes as an image to this post. In particular my main issue comes with (6.9). I can see that at some point he integrates over x to...
  21. baldbrain

    What Are Some Substitutions That Can Be Used to Solve This Integral?

    Let I = ##\int{\sqrt{\frac{cosx - cos^3x} {1-cos^3x}}}\,dx## I = ##\int{\sqrt{\frac{cosx(1 - cos^2x)} {1 - cos^3x}}}\,dx## I = ##\int{\sqrt{\frac {cosx} {1 - cos^3x }}}sinx\,dx## Substitute ##cosx = t## Therefore, ##sinx\,dx = -dt## So, I = ##\int{-\sqrt{\frac {t} {1 - t^3}}}\,dt## I'm stuck...
  22. mishima

    How can the indefinite integral of e^(1/x)/(x(x+1)^2) be solved by hand?

    $$\int \frac {e^{1/x}} {x(x+1)^2} \, dx$$ I came across this indefinite integral when solving a second order differential equation using reduction of order. My CAS can solve it easy enough, but I was wondering what technique could be used to solve it by hand. I have tried some standard...
  23. looseleaf

    I Gaussian Integral Coordinate Change

    Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do...
  24. N

    I Proving a property of an integral

    I'm pondering something about properties of integrals. What can we say about the following limit? ##\lim_{t\to\infty} \int_t^{\infty } f(x) \, dx## On one hand, the 'gap' from the lower to upper integration limit diminishes, so that would suggest the limit is always 0. But what if f is an...
  25. Abhishek11235

    I Changing Summation to Integral

    This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
  26. S

    Analytical solution of the Photon Diffusion Equation

    Homework Statement Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique. Homework Equations Starting equation integrated over t: int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1) Result...
  27. JD_PM

    Understanding the argument of the surface area integral

    Homework Statement Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y## Homework EquationsThe Attempt at a Solution [/B] I have already posted this question on MSE...
  28. JD_PM

    Trying to find an alternative for solving an integral

    Homework Statement Homework Equations The Attempt at a Solution [/B] I understand how the integral is solved using cartesian coordinates. However, I wanted to try to solve it using polar coordinates: $$\int_0^{\pi/2} cos \theta \sqrt{1+r^2 cos^2 \theta}d...
  29. physics bob

    I Solving Quantum Mechanics Integral Equation: How to Get from (1) to (2)?

    The book on quantum mechanics that I was reading says: d<x>/dt = d/dt ∫∞-∞ |ψ(x,t)|2 dx =iħ/2m ∫∞-∞ x∂/∂x [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (1) =-∫∞-∞ [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (2) I want to know how to get from (1) to (2) The book says you use integration by part: ∫abfdg/dx dx = [fg]ab - ∫abdf/df dg dx I chose f...
  30. CCMarie

    A Multi-variable function depending on the Heaviside function

    How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...
  31. astrocytosis

    Volume integral over a gradient (quantum mechanics)

    Homework Statement 1) Calculate the density of states for a free particle in a three dimensional box of linear size L. 2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0## 3) Calculate the integral ##\int...
  32. matai

    Integral for the linear speed of the Earth

    I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...
  33. binbagsss

    Did I Misinterpret the Change of Sign in the Feynman Propagator Integral?

    Homework Statement [/B] Hi in the first attachment I am stuck on the sign change argument used to get from line 2 to 3 , see below Homework Equationsabove The Attempt at a Solution [/B] Q1) please correct me if I'm wrong but : ##d^3 p \neq d\vec{p} ## since ##d^3 p = dp_x dp_y dp_z ## and...
  34. H

    Using hyperbolic substitution to solve an integral

    Homework Statement Homework Equations So the question is asking to solve an integral and to use the answer of that integral to find an additional integral. With part a, I don't have much problem, but then I don't know how to apply the answer from it to part b. I know I should subsitute all...
  35. 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...
  36. C

    Calculating Net Torque in a Rotating System

    Homework Statement hi my question is that say if rod of length l is 3m and is 90 degrees from horizontal pointing upwards with 2 kg weight on the end then there will be no torque produced because the force of gravity is acting parallel to the lever arm if the angle theta is 80 degrees then the...
  37. opus

    How can a series be equal to this integral?

    Homework Statement Determine the series that is equal to the integral ##\int_0^1 x^2\cos(x^3)dx## Homework EquationsThe Attempt at a Solution So I didn't really know what I was doing but I did end up with the correct solution. What I did was to find a Taylor Series for the integrand, this...
  38. JD_PM

    Changing the order of a triple integral

    Homework Statement $$\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) dxdzdy$$ Change the order of the integral to $$\iiint f(x,y,z) \, \mathrm{dydzdx}$$What I have done It is just about: From ##x=0## to ##x=\sqrt{4y+23}## From ##z=0## to ##z=4-y## From ##y=\frac{x^2-23}{4}## to...
  39. JD_PM

    Evaluate an integral over a triangle

    Homework Statement Let D be the triangle with vetrices ##( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 )##. Evaluate the integral : $$\iint_D e^{\frac{y-x}{y+x}}$$ Homework EquationsThe Attempt at a Solution [/B] The answer to this problem is known (...
  40. F

    Flux/Surface Integral across a Plane

    Homework Statement I attached an image of the multi-part problem on this post. I got correct answers to every question other than the last one. Homework EquationsThe Attempt at a Solution I believe the last part is a surface integral problem. F is given and I found n is previous parts of...
  41. V

    B Solving an Integral on a Spherical Surface - Tips

    Hello. I ask for solution help from the integral below, where y and x represent angles in a metric of a spherical, 2-D surface. He was studying how to obtain the geodesic curves on the spherical surface, the sphere of radius r = 1, to simplify. The integral is the end result. It is enough, now...
  42. opus

    I Looking for help to understand the Integral Test

    Please see the attached images that reference the text. To my understanding, we wish to use the integral test to compare to a series to see if the series converges or diverges. In these two examples, we use ##\sum_{n=1}^\infty \frac{1}{n}## compared with ##\int_1^{k+1} \frac{1}{x}dx## and for...
  43. Boltzman Oscillation

    I What is the integral of energy?

    So the classical law of force given by Newton is F= ma = dp/dt = qE. Thus if i integrate the last two equivalents I get: ∫(dp/dt)dt = q∫Edt p + C = q∫Edt correct? then what would the integral of...
  44. A

    Ampere's Law: Double Negative Line Integral Help

    Hi all, I’m having some trouble finding a minus sign in a standard calculation I have been doing. I am trying to show that if there is no enclosed current around the example loop in the enclosed jpeg, the four piecewise paths add up to zero (for the line integral part of Amp’s law). For this...
  45. opus

    Does the series converge using the integral test?

    Homework Statement Use the integral test to compare the series to an appropriate improper integral, then use a comparison test to show the integral converges or diverges and conclude whether the initial series converges or diverges. ##\sum_{n=3}^\infty \frac{n^2+3}{n^{5/2}+n^2+n+1}## Homework...
  46. Othman0111

    Contour Integral to Properly Posting Math Problems on Online Forums

    Homework Statement Homework Equations The Attempt at a Solution
  47. CivilSigma

    Find the PDF in terms of another variable

    Homework Statement For $$f_x(x)=4x^3 ; 0 \leq x \leq 1$$ Find the PDF for $$ Y < y=x^2$$ The Attempt at a Solution So, we take the domain on x to be: $$0\leq x \leq \sqrt y$$ and integrate: $$ \int_0^{\sqrt y} f_x(x) dx = \int_0^{\sqrt y} 4x^3 dx$$ Do we integrate with respect to x or y...
  48. nomadreid

    I |Li(x) - pi(x)| goes to 0 under RH?

    Extremely quick question: According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to |Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c. Am I correct that then c goes to 0 as x goes to infinity? Does any expression exist (yet) for c? Thanks.
  49. Santilopez10

    How Does the Sinc Function Integral Relate to Quantum Collision Theory?

    Homework Statement The following is a problem from "Applied Complex Variables for Scientists and Engineers" It states: The following integral occurs in the quantum theory of collisions: $$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$ where p is real. Show that $$I=\begin{cases}0 &...
  50. CivilSigma

    How to Compute Auto-correlation and Spectral Density of a Damped Sine Wave?

    Homework Statement I am computing the auto correlation and spectral density functions of the following signal: $$f(t)=Ae^{-ct}sin(\omega t)$$ $$AutoCorrelation = R_x(\tau) = \int_{-\infty}^{\infty} f(x)f(x+\tau) \cdot \frac{1}{T} dx$$ $$SpectralDensity = S_x(\omega) = \frac{1}{2\pi}...
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