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. A

    Meaning of a integral equation

    Thanks
  2. C

    MHB Inequality involving Gaussian integral

    I'm trying to solve the inequality: $$ \int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx $$I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there. Any ideas?
  3. benorin

    I Fractional Integral of which function is equal to Riemann's Zeta-Function?

    So the problem I’m attempting to solve is ##\lim_{x\to a} I_{\alpha}f(x)=\zeta (\alpha )## for f, and a, where ##\zeta (\cdot )## is the Riemann zeta function and ##I_{\alpha}## is the Riemann-Liouville left fractional integral operator, namely the integral equation $$\lim_{x\to...
  4. topsquark

    MHB Dimensional regularizatoin of an integral

    This question hopefully isn't going to go too deep into the concept, just a couple of questions to get me going. I am working on using dimensinal regularization of a loop integral in QED. I don't think the specific application to QED is important, but I will say that the original integral is...
  5. L

    Where can I find a long list of clearly solved trig integral problems?

    Homework Statement:: I need to develop my instincts on when to use u-sub, integration-by-parts, trig substitution, etc. But, I need to read/see tons of problems actually being solved with these techniques to know which technique to apply quickly. Relevant Equations:: Sorry for the vague...
  6. acalcstudent

    I Bernoulli Equation with weird integral

    Part of me thinks this is could be a u-sub b/c x^3's derivative is 3x^2, a factor of 3 off from what e is raised to...but it is not a traditional u-sub...any thoughts if this is a u-sub or by parts, and what u should be?I know that there is more to solving the equation after this ( z =...
  7. D

    Represent a 3d region and compute this triple integral

    Let ## E=\left\{ (x,y,z) \in R^3 : 1 \leq x^2+y^2+z^2 \leq 4, 3x^2+3y^2-z^2\leq 0, z\geq0 \right\} ## - Represent the region E in 3-dimensions -represent the section of e in (x,z) plane -compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz## the domain is a sphere of radius 2 with an inner...
  8. S

    Schwarzschild coordinate time integral

    I have tried integration by parts where, ##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*} = \frac{1}{\sqrt{(r*)^3}} \frac{r^{3/2} dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##u = r^{3/2} \quad \quad dv = \frac{dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##du = \frac{3}{2} r^{1/2} dr...
  9. K

    MHB Troubleshooting a Trigonometric Integral: Algebra and Solutions

    I have a few questions and a request for an explanation. I worked this problem for a quite a while last night. I posted it here. https://math.stackexchange.com/questions/3547225/help-with-trig-sub-integral/3547229#3547229 The original problem is in the top left. Sorry that the negative...
  10. L

    Integral for the calculation of torque

    Hello, I found an integral to calculate the torque from the applied torsional shear stress, and I didn't find an explanation of how this integral is deviated. Where does it come from? Could someone explain? T = ∫τ⋅r⋅dA = ∫τ⋅2πr⋅dr, where T is the torque and τ the shear stress. Thanks a lot!
  11. archaic

    Trigonometric definite integral of 1/(4-sqrt(x))

    This could be solved by the substitution ##u=\sqrt x##, but I wanted to do it using a trigonometric one. The answer is false, but I don't see the wrong step. Thank you for your time! [Poster has been reminded to learn to post their work using LaTeX]
  12. D

    Does it look like I'm doing this double integral correctly?

    are the boundaries of integration correct? i split the domain in two as follows -2<=x<=0 , -(4-x^2)^(1/2)<y<=x+2 and 0<=x<=2 -(4-x^2)^(1/2)<=y<=(4-x^2)^(1/2)
  13. P

    I An integral rewritten (from “Almost impossible integrals“, p.59 in Valean)

    I want to understand where the minus 1 in the first line in the RHS term comes from. I assume the little apostrophe means taking a derivative. But the antiderivative of x^(n-1) is (1/n)x^n. Why the -1? thank you
  14. K

    I Why the integral of a complex exponential can't be equal to zero?

    I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero. On one hand, as we can in principle choose whatever values we like for ##m##...
  15. Yohan

    Finding if an improper integral is Convergent

    find out for what values of p > 0 this integral is convergent ##\displaystyle{\int_0^\infty x^{p-1}e^{-x}\,dx}\;## so i broke them up to 2 integrals one from 0 to 1 and the other from 1 to ∞ and use the limit convergence test. but i found out that there are no vaules of p that makes both of...
  16. benorin

    I An Improper Multiple Integral

    The Lerch Transcendent identity from my paper which may or may not be true, for ##N\in\mathbb{Z}^+##, and I forget the domain of z and y, here it goes $$\Phi (z,N,y) :=\sum_{q=0}^{\infty}\frac{z^q}{(q+y)^N}$$ $$=\int_{0}^{1}\int_{0}^{1}\cdots \int_{0}^{1}\prod_{k=1}^{N}\left(...
  17. topsquark

    MHB Regularization of an Non-conergent Integral

    The equations here come from calculating the amplitude of a Feynman diagram. I can set up the problem if you really want me to but here I am just interested in why and how the regularization process is supposed to work Mathematically. The generalized meaning of this is if we are given a...
  18. D

    Double integral domain with absolute value

    D={(x,y)∈ℝ2: 2|y|-2≤|x|≤½|y|+1} I am struggling on finding the domain of such function my attempt : first system \begin{cases} x≥2y-2\\ -x≥2y-2\\ x≥-2y-2\\ -x≥-2y-2 \end{cases} second system \begin{cases} x≤y/2+1\\ x≤-y/2+1\\ -x≤y/2+1\\ -x≤-y/2+1\\ \end{cases} i draw the graph and get the...
  19. Adesh

    Why does dividing by ##\sin^2 x## solve the integral?

    If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...
  20. M

    I Stokes Theorem: Vector Integral Identity Proof

    Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...
  21. D

    I Question about this double integral

    could please some one explain the inequality on the right? in particular how should i see and thanks
  22. I

    Integral of relative distance–dependent potential

    I think its going to be intg(dr2)intg(exp(r^2) dr) or something like that.
  23. A

    Expressing an Integral as a sum of terms

    e.g Can we write it as $$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+...f(b)=\int^b_a f(x)dx$$...(?) Although $$\int f(x)dx$$ given the area tracked by thr function with the x-axis between a and b Thanks.
  24. R

    I Double integral and Green's theorem

    Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region
  25. R

    Line integral around a circle centered at the origin

    Hi everyone, I am confused in this question. First I solved it by noticing that the gradient of the function will be zero (without substitution the hit) I got that it's a conservative field so the integral should be zero since it's closed path. Then I solved it by the hit and convert it as any...
  26. R

    Change of variable in a double integral

    Hi everyone, I tried to solve the last part of the question, I substituted back the expression of x and y into the equation of the ellipse, I got that r=1 or r=-1. But got no idea how to find the boundary for theta, I got a guess that, It should be from zero to pi. But got no reason why to...
  27. Math Amateur

    MHB The Riemann Integral .... Conway, Proposition 3.1.4 ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 3: Integration ... and in particular I am focused on Section 3.1: The Riemann Integral ... I need help with an aspect of the proof of Proposition 3.1.4 ...Proposition 3.1.4 and its proof read as follows...
  28. karush

    MHB 3.1.6 AP calculus Exam piece wise integral

    I tried to do this just by observation, but kinda hard with a piece wise function so would presume
  29. A

    Double integral with polar coordinates

    Hello there, I'm struggling in this problem because i think i can't find the right ##\theta## or ##r## Here's my work: ##\pi/4\leq\theta\leq\pi/2## and ##0\leq r\leq 2\sin\theta## So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta## Which is equal to...
  30. tworitdash

    A Integral of 2 Bessel functions of different orders

    I can only find a solution to \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel...
  31. WMDhamnekar

    MHB Differentiation under integral sign

    Hello, How to find formulas for these$\displaystyle\int x^n\sin(x)\, dx, \displaystyle\int x^n\cos(x)\, dx,$ indefinite integrals when $n=1,2,3,4$ using differentiation under the integral sign starting with the formulas $$\displaystyle\int \cos(tx)\,dx = \frac{\sin(tx)}{t}...
  32. G

    MHB Solve integral with laplace transform

    So the task is to solve the following integral with laplace transform. Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)). What I am unsure about is what happens with the integral part and how do we inpret the resulting expression? What will it result...
  33. W

    A Doubt about Energy Condition in Wormhole: Integral Along Null Geodesic

    I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$ And I apply the worm hole space...
  34. karush

    MHB 1.8.4 AP Calculus Exam Integral of piece-wise function

    image due to macros in Overleaf ok I think (a) could just be done by observation by just adding up obvious areas but (b) and (c) are a litte ? sorry had to post this before the lab closes
  35. karush

    MHB 15.1.34 Evaluate triple integral

    15.1.34 Evaluate $\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}} \sin{y} \, dz \, dx \, dy$ integrat dz $\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy $ integrat dx $\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$...
  36. E

    B The use of the dx in the quantum state vector integral

    As a simple example, the probability of measuring the position between x and x + dx is |\psi(x)|^{2} dx since |\psi(x)|^{2} is the probability density. So summing |\psi(x)|^{2} dx between any two points within the boundaries yields the required probability. The integral I'm confused about is...
  37. D

    Solving this integral in 2 different ways

    The answer gives $$ \int x /(2x-1)\ dx = x/2 +(1/4)ln|2x-1| + C $$ whicjh I can obtain. But when I try a different way I get a different answer. I must be making a stupid mistake but I can't see it. Here is my method $$ \int x/(2x-1) dx = \int x/[2(x-(1/2)] dx = (1/2) \int x/(x-(1/2)) dx $$...
  38. dRic2

    Evaluate the following integral from a physics textbook

    Using spherical coordinates I can write ##d^3 k = 2\pi k^2 \sin \phi dk d \phi## (where I've already preformed the integration along the azimuthal angle, yielding the factor ##2 \pi##). Btw I'm sorry for my unfortunate notation: usually ##\phi## is the azimuthal angle, but here it is the polar...
  39. 0

    Integral of 1/ln(x). Convergence test

    Some functions have straight foward integrals, but they get complicated if you take the inverse of it. 1/f(x) for instance. The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges. ##\int_1^\infty (\ln(x))^n dx## If n = 0, I...
  40. F

    I When we can change a sum to an integral?

    In physics we often change a sum to an integral.But I am not clear when can we change a sum to an integral?When a term of sum is comparable to the sum,can we change the sum to integral?
  41. M

    I Integral in a variational principle problem

    Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't...
  42. M

    Find the volume of the solid formed by the rotation around the y=0

    Hi, I find this... Please tell me your opinion on this. Thanks.
  43. M

    What's the integral of a unit vector?

    So I'm trying to figure out the integral of phi hat with respect to phi in cylindrical coordinates. My assumption was that the unit vector would just pass through my integral... is that correct? (I reached this point in life without ever thinking about how vectors go through integrals, and...
  44. Saracen Rue

    I Integral involving up-arrow notation

    I was playing around with a graphing program and sketching polar graphs involving tall power towers, when I noticed that ##sin(\theta) \uparrow \uparrow a## has an alternating appearance depending on whether ##a## is odd or even. I also noticed that the area enclosed by these alternating graphs...
  45. K

    I Is a Vector Field Equal to Zero if Its Contour Integral is Zero?

    I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?
  46. D

    Integration by Parts for Complex Integrals

    integral of x^12 sinx dx = x^12 -cosx - 12x^11 sinx - cosx (132x^11/11)
  47. RicardoMP

    Feynman one-loop integral ##I_{21}##

    Starting from the general formula: $$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\epsilon}{2})}{\Gamma(2-\frac{\epsilon}{2})\Gamma(n)}\frac{1}{\Delta^{n-m-2}}(\frac{4\pi M^2}{\Delta})^{\frac{\epsilon}{2}}\Gamma(n-m-2+\frac{\epsilon}{2})$$ I arrived to the following...
  48. P

    Expressing this vector integral as a tensor involving the quadrupole

    Before writing out each component I'm going to simplify ##\vec{I}## to the best of my abilities $$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$ $$\vec{I} = \hat{r} \cdot \int \vec{r'} \left( x' , y', z' \right) \rho\left( \vec{r'} \right)\...
  49. jisbon

    Showing that integral is an induction

    Hi all, Having this equation derived: ##\int _0^{\frac{\pi }{2}}\:sin^{n}x\:dx\:=\:\frac{n-1}{n}\int _0^{\frac{\pi }{2}}\:sin^{n-2}x\:dx## What I will do is simply substitue n with n+2, and I will get the following: ##\frac{2n}{2n+1}\int_{0}^{\pi /2}(sinx)^{2n-1}dx## What should I do from here?
  50. H

    A Double Integral with Dirac Delta Function and Changing Limits

    I have an integral: \int_{-1}^{0}\int_{-1}^{q}\delta(s+a)\sinh[k(q-s)]dsdq where 0<a<1 and \delta (s-a) is a dirac delta function. Anyone know what to do?
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