Integrals Definition and 1000 Threads

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. T

    Volume, surface, and line integrals

    Homework Statement Consider a vector A = (x^2 - y^2)(i) + xyz(j) - (x + y + z)k and a cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1 Determine the volume integral ∫∇.A dV where V is the volume of the cube Determine the surface integral ∫A.n dS where s is the surface of...
  2. alyafey22

    MHB Prove the Legendre equation on Elliptic integrals

    K(k)E'(k)+K'(k)E(k)-K(k)K'(k)=\frac{\pi}{2} Complete Elliptic integral of first kind K(k)= \int^1_0 \frac{dx}{\sqrt{1-x^2}\sqrt{1-k^2x^2}} Complete Elliptic integral of second kind E(k)= \int^1_0 \frac{\sqrt{1-x^2}}{\sqrt{1-k^2x^2}}dx Complementary integral...
  3. B

    Where Can I Practice Integrals Using Infinite Series?

    On the bottom of page 24 & top of page 25 of this pdf an integral is beautifully computed by breaking it up into an infinite series. Is there any reference where I could get practice in working integrals like these?
  4. V

    MATLAB Maximizing the fraction of two integrals using matlab

    EDIT: I left out something of major importance, I want to maximize with respect to a! Good day, I've run into an issue in solving a certain problem with matlab, and I was hoping if anyone could help me out. I am relatively new to matlab, so I don't even know if this is possible, but I...
  5. V

    Maximizing the fraction of two integrals

    EDIT: I left out something of major importance, I want to maximize with respect to a! Homework Statement My problem is rather complex, but in the end it boils down to maximizing the fraction Homework Equations With the calculus I know, I can't evaluate these integrals, so I have no applicable...
  6. M

    Rewriting sum of iterated integrals (order of integration)

    Homework Statement Rewrite the given sum of iterated integrals as a single iterated integral by reversing the order of integration, and evaluate. $$\int_0^1 \int_0^x sin x dy dx + \int_1^2 \int_0^{2 - x} sin x dy dx$$ Homework Equations None The Attempt at a Solution I drew the domains of...
  7. L

    Dirac Delta Integrals: How to Solve for the Argument of the Delta Function?

    Homework Statement This is just an example, not a specific problem. So if I have ∫σ(sinx), for example, and my limits of integration are, for example, 1 to 10, what I need to do to solve that is to find a value of x that would make the argument of the delta function 0. So for sinx, 0 makes...
  8. M

    Double Integrals: Computing P with Constant Limits

    Hi all, I have a question regarding certain double integrals. Assume the function $$ l(t) $$ is given as well as the function $$K(t)$$, defined only for positive argument. Also the definition $$n(t) = \int_{-\infty}^{t} K(t-\tau) l(\tau) \mathrm{d}\tau$$ is given. if I wish to compute $$P =...
  9. C

    Integrals in cylindrical coordinates.

    Integrate the function f(x,y,z)=−7x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt(263/137)x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20. I...
  10. J

    MHB Integrating Trigonometric Functions with Multiple Substitutions

    [1] $\displaystyle \int\sqrt{\frac{\csc x-\cot x}{\csc x+\cot x}}\cdot \frac{\sec x}{\sqrt{1+2\sec x}}dx$ [2] $\displaystyle \int \frac{3\cot 3x - \cot x}{\tan x-3 \tan 3x}dx$ Thanks pranav I have edited it.
  11. DreamWeaver

    MHB A generalized Clausen Function, and associated loggamma integrals

    I've recently been working on a number of integrals related to the loggamma function, so I thought I'd share my results here. I'll have to post as and when I have time, and there will be a fair bit of preliminary work before we get to the final results, but - loosely speaking - the main aim here...
  12. H

    Coordinate transformation for line integrals; quadrature rules

    Hi all, The context of this problem is as follows: I'm trying to implement a discontinuous finite element method and the formulation calls for the computation of line integrals over the edges of the mesh. Anyway, more generally, I need to evaluate \int_{e}f(x,y)ds, where e is a line segment...
  13. W

    When Should Integrals Be Applied in Physics Problems?

    Homework Statement A crate of mass 9.6 kg is pulled up a rough incline with an initial speed of 1.44 m/s. The pulling force is 92 N parallel to the incline, which makes an angle of 19.4° with the horizontal. The coefficient of kinetic friction is 0.400, and the crate is pulled 4.92 m. (d)...
  14. B

    MHB Difficult Improper Integrals in Real Analysis.

    Hello. I'm studying improper integrals in real analysis. However, two problems are very difficult to me. If you are OK, please help me.(heart) 1.2. I have solutions about above problems. However, I don't know how I approach and find the way for solving them.
  15. R

    Convergence of improper integrals theorems

    Homework Statement I'm trying to prove these two theorems a) if ## 0 \leq f(x) \leq g(x) ## for all x ## \geq 0 ## and ## \int_0^\infty g ## converges, then ## \int_0^\infty f ## converges b) if ## \int_0^\infty |f| ## converges then ## \int_0^\infty f ## converges. Obviously assuming...
  16. J

    MHB How to Solve These Two Indefinite Integrals?

    $(a)\;\;:: \displaystyle \int\frac{1}{\left(x+\sqrt{x\cdot (x+1)}\right)^2}dx$ $(b)\;\;::\displaystyle \int\frac{1}{(x^4-1)^2}dx$ My Trial :: (a) $\displaystyle \int\frac{1}{(x+\sqrt{x\cdot (x+1)})^2}dx$ $\displaystyle \int\frac{1}{x\left(\sqrt{x}+\sqrt{x+1}\right)^2}dx =...
  17. T

    Understanding Vector Integral Notation

    Given, \sigma_{b} = \vec{P}\bullet\hat{n} Now, integrate both sides over a closed surface, \oint \sigma_{b} da = \oint (\vec{P}\bullet\hat{n}) da My math is fuzzy, and I don't really understand this next step. \oint \sigma_{b} da = \oint \vec{P} \bullet d\vec{a}...
  18. L

    Gaussian Integrals for Quantum States of well Defined Momentum

    Consider the Gaussian Integral (eqn 2.64).. is anyone able to explain how the constant of normalization is rationalised?
  19. J

    Contour Integration used to solved real Integrals

    Homework Statement Essential Mathematical Methods for the Physics Sciences Problem 15.7 Show that if f(z) has a simple zero at z0 then 1/ f(z) has a residue of 1/f'(z0). Then use this information to evaluate: ∫ sinθ/(a- sinθ) dθ, where the integral goes from -∏ to ∏. Homework...
  20. U

    Find this integral in terms of the given integrals

    Homework Statement If \displaystyle \int_0^1 \dfrac{e^t}{t+1} dt = a then \displaystyle \int_{b-1}^b \dfrac{e^{-t}}{t-b-1} dt is equal to Homework Equations The Attempt at a Solution I used the definite integral property in the second integral \displaystyle \int_{b-1}^b...
  21. S

    Integrals featuring the laplacian and a tensor

    Ok, so I'd like some advice on doing integrals that involve a laplacian and a tensor for example =\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial x^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho}) where...
  22. M

    Green's theorem, relation between two integrals

    Homework Statement . Calculate by a line integral the following double integral: ##\iint\limits_D (y^{2}e^{xy}-x^{2}e^{xy})dxdy##, D being the unit disk. The attempt at a solution. Well, if we consider C to be the curve that encloses the region D (C is the unit circle), then C is a...
  23. H

    Does the left side of Euler's Equation always equal zero?

    Sorry, the title doesn't match up 100% with the content of the topic, but that's because I've decided to be a little bit more explicit about my question. I am trying to walk through the proof of Euler's Equation from Calculus of Variations, and I'm a little bit confused by the final step...
  24. paulmdrdo1

    MHB Is My Solution to This Integral Correct?

    again, i need some help here guys.$\displaystyle\int\frac{3x-1}{2x^2+2x+3}dx$ =$\displaystyle\int\frac{3x-1}{2\left[\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\right]}dx$ $\displaystyle a=\frac{\sqrt{5}}{2}$; $\displaystyle u=x+\frac{1}{2}$; $\displaystyle du=dx$; $\displaystyle x=u-\frac{1}{2}$...
  25. paulmdrdo1

    MHB Are These Trigonometric Integral Solutions Equivalent?

    i was thinking hard how to integrate this, but none of the techniques I know did work. please kindly help with this matter. thanks! $\displaystyle\int\sqrt{1+\cos\theta}d\theta$
  26. C

    Solving a Problem with Integrals in R3

    Hi! I've got a problem with an integral. Let's assume we've got something like this: ∫R3d3x1∫R3d3x2∫R3d3x3∫R3d3x4P(|x1|)P(|x3|)δ(x1+x2)δ(x3+x4)W(|x1+x2|)W(|x3+x4|) xi is a vector The "δ" is the Dirac delta. P(|x|i) & W(|xi+xj|) are some functions I would like to make it looks a bit...
  27. paulmdrdo1

    MHB So I get the same answer as the book except for a constant term.

    i'm kind of unsure of my solution here please check. $\displaystyle\int \sin^3x\cos^3x= \int(\sin x \cos x)^3 =\int(\frac{1}{2}\sin2x)^3=\frac{1}{8}\int \sin^3 2x dx$ $\displaystyle u=2x$; $\displaystyle du=2dx$; $\displaystyle dx=\frac{1}{2}du$ $\displaystyle \frac{1}{8}\int...
  28. T

    Introducing parameters in integrals

    Homework Statement Using the integral ∫dx/1+x^2 = pi/2 from 0 to infinity as a guide, introduce a parameter and then differentiate with respect to this parameter to evaluate the integral ∫dx/(x^2+a^2)^3 from 0 to infinity Homework Equations The Attempt at a Solution ∫(1/1+x^2) =...
  29. skate_nerd

    MHB Non-continuous integrals and discrete variables

    Quantum Phys Homework: I am given a function: $$f(x)=\frac{1}{10}(10-x)^2\,;\,0\leq{x}\leq{10}$$ and $$f(x)=0$$ for all other \(x\). I need to find the average value of \(x\) where $$\bar{x}=\frac{\int_{-\infty}^{\infty}x\,f(x)\,dx}{\int_{-\infty}^{\infty}f(x)\,dx}$$ I am not really even sure...
  30. N

    MHB Residue theorem to evaluate integrals

    Please refer to attached material. For the first question, I have tried looking at examples and have noted that the bounds have been provided in a manner: like |z|=1 (as given in part ii) I am not sure how to get transform the given |z-pi|=pi in such a format, although i suspect it would be...
  31. polygamma

    MHB Log-sine and log-cosine integrals

    For a few of you, this probably isn't very challenging. But I'm going to post it anyways since I find it interesting. Show that for $0 \le \theta \le \pi$, $ \displaystyle \int_{0}^{\theta} \ln(\sin x) \ dx = - \theta \ln 2 - \frac{1}{2} \sum_{n=1}^{\infty} \frac{\sin (2n \theta)}{n^{2}}$.Also...
  32. DreamWeaver

    MHB Logarithmic Integrals, Polylogarithms, and associated functions

    This is not so much a tutorial, but rather a collection of useful results and techniques. Some of the proofs will be quite long, since as much as possible, I'll aim to prove most results and functional relations used herein, rather than just present another's identity as fact. There will be a...
  33. J

    Can the Average Value of an Integral Be Negative?

    Homework Statement I have a question can the average value for an integral be negative. I don't see why not just checking. You know this evalutation f_ave = (1/b-a) ∫ f(x) dx Homework Equations thx The Attempt at a Solution
  34. DreamWeaver

    MHB Inverse Sine/Tangent Integrals and related functions

    Within certain branches of analysis - both real and complex - the Inverse Tangent Integral (and its generalizations) can be quite useful. Similarly, it's much less well-known (= uglier? lol) cousin, the Inverse Sine Integral can be used to solve many problems. To that end, this is not really a...
  35. DreamWeaver

    MHB A few tricky integrals.... Or are they?

    Here are a few Vardi-type integrals I recently posted on another forum (some of you might have seen them)...Assuming the following classic result - due to Vardi - holds...\int_{\pi/4}^{\pi/2}\log\log(\tan x)\,dx=\frac{\pi}{2}\log\left[\sqrt{2\pi}\frac{\Gamma(3/4)}{\Gamma(1/4)}\right]Prove that...
  36. B

    Surface Integrals in Gauss's Theorem of Charge in Motion

    gauss's theorem is also applicable to charge in motion.but how the surface integral has to be taken??
  37. W

    ML-inequality, Estimation of Line Integrals

    Problem: Let ##\vec{F}## be a vector function defined on a curve C. Let ##|\vec{F}|## be bounded, say, ##|\vec{F}| ≤ M## on C, where ##M## is some positive number. Show that ##|\int\limits_C\ \vec{F} \cdot d\vec{r}| ≤ ML ## (L=Length of C).Attempt at a Solution: I honestly have no idea where...
  38. A

    Does triple integrals have to have a specific interval?

    I hope this makes my question clear... suppose we have a triple integral of dzdydx for [0<x<1 , sqt(x)<y<1 , 0<z<1-y] and from the sketch we can see that 0<y<1 and 0<z<1... my question is this, if we change the integration to dzdxdy we get [0<x<y^2 , 0<y<1 , 0<z<1-y], is that the only way? or...
  39. Y

    MHB Mean value theorem for integrals

    Hello all, I have a couple of questions. First, about the mean value theorem for integrals. I don't get it. The theorem say that if f(x) is continuous in [a,b] then there exist a point c in [a,b] such that \[\int_{a}^{b}f(x)dx=f(c)\cdot (b-a)\] Now, I understand what it means (I think), but...
  40. J

    Using laplace transforms to solve integrals

    Homework Statement ##\int_0^\infty \frac{a}{a^2+x^2} dx## Homework Equations All the basic integration techniques. The Attempt at a Solution So, I saw this problem and wanted to try it using a different method then substitution, which can obviously solve it pretty easy. Since it is a very...
  41. O

    Generalization of Mean Value Theorem for Integrals Needed

    Hi all, I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't. Is the following true? If F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n}...
  42. S

    Direct Comparison Test - Improper Integrals

    1. Homework Statement [/b] Use the direct comparison test to show that the following are convergent: (a)\int_1^∞ \frac{cos x\,dx}{x^2} I don't know how to choose a smaller function that converges similar to the one above. The main problem is i don't know where to start. A simple...
  43. T

    Bessel Functions as Solutions to Scattering Integrals?

    Hello All. I'm currently in a crash course on X-ray Diffraction and Scattering Theory, and I've reached a point where I have to learn about Bessel Functions, and how they can be used as solutions to integrals of certain functions which have no solution. Or at least, that's as much as I...
  44. T

    Application of Residue Theorem to Definite Integrals (Logarithm)

    I've been studying for a test and have been powering through the recommended problems and have stumbled upon a problem I just can't seem to figure out. $$\int_{0}^{\infty} \frac{logx}{1+x^{2}} dx$$ (Complex Variables, 2nd edition by Stephen D. Fisher; Exercise 17, Section 2.6; pg. 167)...
  45. A

    Odd Function Integrals: Exploring Answers and Assumptions

    Hello I'd first like to state I know how to solve and I know the answer to this integral however when I first looked at the integral my initial thought was that it was equal to zero. I'd like to explain why I thought it was equal to 0 and hopefully someone can tell me where I went wrong. I...
  46. W

    Weird ways of doing closed loop integrals

    I was looking at an example where it was evaluating a closed loop integral of a vector field around a triangle (0,0) (1,1) (2,0) by using greens theorem. This example was in the green's theorem section of the book so green's theorem must be used. Anyways the double integral was set up as follows...
  47. Y

    Calculate definite integrals with given interval.

    I just want to verify is this the way to calculate the result of a definite integral with the given interval. Say the result of the integral over [0,##\frac{\pi}{2}##] is \sin(\theta)\cos(\theta)d\theta|_0^{\frac{\pi}{2}} It should be...
  48. S

    Boundaries for double integrals?

    And your boundaries are defined as: 0 < x < y < 1 How do you know the relationship between x and beyond this? That is, we know that y is between x and 1, but x is between 0 and y. We have a loop. In a specific example, I know the answer is, where f(x,y) = 8xy ∫∫8xy dx dy With bounds 0 to...
  49. J

    Learning Gaussian Integrals for Quantum Mechanics

    Good evening Im starting to learn quantum mechanics from Griffith's book however I am having problems when dealing with Gaussian integrals in the first chapter. What book should I read in order to understand this subject? are there resources about gaussian integrals out there? Thanks a lot.
  50. mathworker

    MHB Integral Evaluation: x^2+4 & 2+2sinx+cosx

    Evaluate integrals: 1) \int\frac{dx}{x^2\sqrt{x^2+4}} 2) \int\frac{dx}{2+2\text{sin}x+\text{cos}x}
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