What is Integrals: 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. MAGNIBORO

    I Question about Complex limits of definite integrals

    Hi, I see a formula of gamma function and i have a question. (1) $$\Gamma (s) = \int_{0}^{\infty } e^{-x}\, x^{s-1} dx$$ (2) $$ x=a\, n^{p} \rightarrow dx=ap\, n^{p-1}dn$$ (3) $$\frac{\Gamma (s)}{pa^{s}} = \int_{0}^{\infty } e^{-an^{p}}\, n^{ps-1} dn$$ i understand the formula but...
  2. S

    Integrals with infinite well eigenfunctions

    Homework Statement This is problem 17 from Chapter 3 of Quantum Physics by S. Gasiorowicz "Consider the eigenfunctions for a box with sides at x = +/- a. Without working out the integral, prove that the expectation value of the quantity x^2 p^3 + 3 x p^3 x + p^3 x^2 vanishes for all the...
  3. bwest121

    How do I calculate this integral?

    Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...
  4. J

    I Question about definite and indefinite integrals

    First, just to check, I write what I think and let me know if I am wrong: The definite integral of a function gives us a number whose geometric meaning is the area under the curve between two limiting points. We can calculate this integral as the limit of the sum of the rectangles and the...
  5. A

    Substitution Rule for Integrals: How to Simplify Complex Integrands

    Homework Statement $$\int_{0}^{2} r\sqrt{5-\sqrt{4-r^2}} dr$$ Homework EquationsThe Attempt at a Solution would i substitute ##u=4-r^2##? After of which I would input into the integral and get: $$\int_{0}^{2} \sqrt{5-\sqrt{u}}du$$ What would I do here? Do I just work inside the radical(so 5r...
  6. A

    Trigonometric functions and integrals

    Homework Statement I'm searching for the integral that gives arcosu Homework Equations as we know : ∫u'/[1-u^2]^0.5 dx = arcsinu derivative of arccosu = -u'/[1-u^2]^0.5 + C derivative of arcsinu= u'/[1-u^2]^0.5 The Attempt at a Solution when I type the -u'/[1-u^2]^0.5 on the online integral...
  7. D

    I A question about momentum integrals and lengths

    I've been making my way through Matthew Schwartz's QFT book "Quantum Field Theory and the Standard Model". In chapter 6 he derives the differential cross-section for a ##2\rightarrow n## interaction. As part of the derivation, he introduces the Lorentz invariant phase space measure (LIPS), and...
  8. A

    Equating two integrals with a constant involved

    Homework Statement ##\int_{-1}^{3} f(x) dx = -4 = \int_{-1}^{3} 2g(x)dx## Now find a value(constant a) that makes the following true: ##\int_{-1}^{3} [3f(x) - ag(x) +a] dx = \int_{-1}^{3}(1-ax)dx## Homework EquationsThe Attempt at a Solution I'm unsure if my approach here is correct but I...
  9. S

    I Spivak's Definition of Integrals vs Stewart's

    Hello. I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my...
  10. toforfiltum

    Calculating work done using line integrals

    Homework Statement Sisyphus is pushing a boulder up a 100-ft tall spiral staircase surrounding a cylindrical castle tower. a) Suppose Sisyphus's path is described parametrically as $$x(t)=(5\cos3t, 5\sin3t, 10t)$$, $$\space 0\leq t\leq10$$. If he exerts a force with constant magnitude of 50 Ib...
  11. S

    I Successive Integrals: Examining 1st Integral y_1(t)

    My textbook shows the following example of successive integrals: Does the 1st integral y_1(t) have a typo? Shouldn't the integral be taken with regards to Tau? If not, I would appreciate an explanation of why the 1st successive integral is taken with regards to t, but in the second it is taken...
  12. O

    Integral of Sqrt(x)*e^-x: Step-by-Step Solution

    Homework Statement Evaluate the following integral: ∫0∞ √(x)* e-x dx Homework Equations ∫0∞ e-x2 dx = (√π)/2 The Attempt at a Solution So far this is what I've done: u = x1/2 du = 1/2 x-1/2 2 ∫ e-u2 u2 du Now, I'm not really sure what to do? Or if what I've done so far is leading me down...
  13. N

    MHB Evaluating Integrals: ∫_(-4)^1[f(x)]dx = (-125)/6

    Can anyone tell me if this is correct? \int ∫1 at top and -4 on bottom of symbol [1^3/3+3 1^2/2-(4 x 1)+C] - [〖-4〗^3/3+〖3x(-4)〗^2/2-(4 x-4)+C] If f(x) = x^2+3x-4, then F(x) = x^3/3+3 x^2/2-4x+C ∫_(-4)^1[1^3/3+3 1^2/2-(4 x 1)+C] - [〖-4〗^3/3+〖3x(-4)〗^2/2-(4 x-4)+C] [1/3+3/2-4+C] - [-64/3+24+16+C]...
  14. Stephanus

    B Learning Integrals with the Graph of y=x^2

    Dear PF Forum, I'd like to study integral. But I just realize that I'm lack of basic integral. Can I ask here? Let y=x2 Here is the graph. So, 1. dx is the distance between the red vertical lines? But it's very, very, very small distance. 2. f(x) * dx is the yellow area? 3. ##\int_0^2...
  15. S

    A Line integrals of differential forms

    Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a...
  16. Hamal_Arietis

    Calculating Definite Integrals of $$f_n(x)$$

    Homework Statement Given the function in x $$f_n(x)=sin^nx (n=1,2,3,...)$$ For this ##f_n(x)##, consider the definite intergral $$I_n=\int_{0}^{\pi/2}f_n(x)sin2xdx$$ a) Find ##I_n## b) Hence the obtain $$lim_{n→∞}(I_{n-1}+I_n+I_{n+1}+...+I_{2n-2})=\int_0^W\frac{X}{Y+x}dx$$ Find X,Y,Z. Homework...
  17. nysnacc

    Tackling Difficult Homework Integrals

    Homework Statement Homework Equations ∫ F dt = mv The Attempt at a Solution I have ∫ T -mg dt = ∫ 100+2t2 - mg dt = ∫ 100-9000*9.81 +2t2 dt but then I got negative impulse ...??
  18. S

    A Convergence properties of integrals

    It is perfectly fine to do the following: ##\displaystyle{\int_{-\infty}^{\infty}\ d\phi\ e^{-\phi^{2}/2}e^{-\lambda \phi^{4}/4!} = \int_{-\infty}^{\infty}e^{-\phi^{2}/2}\sum\limits_{n=0}^{\infty}}\frac{(-\lambda\phi^{4})^{n}}{(4!)^{n}\...
  19. F

    I Definitions of the Riemann integral

    In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...
  20. A

    Calculus of Variations: Functional is product of 2 integrals

    Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...
  21. Pull and Twist

    MHB ANSWER CHECK: Sum of Double Integrals involving Polar Conversion

    Here is the given problem... And I first approached it by drawing the xy footprint to get my theta and radius limits to convert to polar. Then I overlooked the original xy function and pretty much took the area of that footprint (highlighted in green.) That gave me a very nice number...
  22. superkraken

    Surface integrals and line integrals

    Homework Statement when we calculate the electric field due to a plane sheet or the magnetic field due to a wire,are we calculating it at a single point or the whole field due to the total wire? Homework EquationsThe Attempt at a Solution
  23. K

    I Green's theorem and Line Integrals

    (Sorry for my bad English.) I was reading about the Green's theorem and I notice that the book only shows for the case where the function is a vector function. When learning about line integrals, I saw that we can do line integrals using "ordinary" functions. For example, the line integral of...
  24. N

    Double integrals with variable upper limit

    How can you compute F(k) = k\int^{\infty}_{0}dy\int^{y}_{0}dx f(kx,y) in C. I know about Python's scipy.integrate.dblquad function but it's just too slow. I have written some Cython code with a 2D gaussian quadrature function in C but it only takes doubles as limits. I think C doesn't have...
  25. micromass

    Insights Some Misconceptions about Indefinite Integrals - Comments

    micromass submitted a new PF Insights post Some Misconceptions on Indefinite Integrals Continue reading the Original PF Insights Post.
  26. J

    MHB Possible title: Troubleshooting File Attachments on Online Forums

    a) The two surfaces defined by xz^2 + 2yz +xy^2 = 4, 2x - 7y - 3z = 2 intersect in a curve S. Find the tangent line and normal plane to S at the point (3,1,-1). b) Sketch the graphs of y = x^m and y = x^n where m and n are...
  27. L

    A Saddle point Integrals with logarithm and cosine

    I'm trying to use the saddle point method to solve the following integral: Z = (1/sqrt{2 pi t}) ∫_{1}^{infinity} ds (1/sqrt{2 pi s}) exp{ p [-s ln(s/t) +s] } cos(2 pi L~ p ~ s), as p → infinity Mod edit to make integral more readable: $$Z = \frac{1}{\sqrt{2\pi t}} \int_1^{\infty} \frac 1...
  28. N

    Solve Integral: y'=x^2, Point (2,6) on Graph

    Homework Statement Determine the function for which y '= x ^ 2, and (2,6) is a point of the graph. Homework Equations y(x)=∫ x^2 dx The Attempt at a Solution I tried doing this but didn't get the right answer y(x)=∫ x^2 dx = x^3/3 +c x=2 y=6 6=(2)^3 /3 +c C= 3,33... But according to...
  29. JuanC97

    I Why Does Integrating |f(x)| Differ from Integrating f(x)?

    I know that \sqrt{f(x)^2} = |f(x)| However... I've just noticed that integrals of expressions like this are usually assumed to be equal to the integral of f(x) without the absolute value. I'd like to know how that's possible. Is weird for me to consider those expressions; specially because of...
  30. A

    How do integrals actually work?

    Homework Statement Calculate the electrical energy required to assemble a spherical volume of radius R and charge Q, homogeneous density ρ the answer is (3/5)Q/R the textbook says you have to build the volume integral one layer of sphere at a time, I'll get back to that later. I like...
  31. L

    A Exploring Gamma Functions: Analytic Possibilities & Integrals

    I have two questions related Gamma functions 1. Finding ##\Gamma## analytically. Is that possible only for integers and halfintegers? Or is it possible mayble for some other numbers? For example is it possible to find analytically ##\Gamma(\frac{3}{4})##? 2. Integral...
  32. P

    Find, using complex contour integrals, the function f(x)....

    Homework Statement whose Fourier transform is f~(p) = 1/(a2 + p2) Homework Equations f(x) = 1/√2π ∫∞-∞ eipx f~(p)The Attempt at a Solution First of all I let f(z) = eixz/(z2 + a2) and γ = γ1 + γ2 with the ϒ's parametrised by: γ1 : {z=t, -R<t<R} γ2 : {z=Reit, 0<t<π} (So a semicircle of radius...
  33. Seanra

    I The speed of light in a medium and path integrals

    So I've heard from multiple sources that one explanation for why light slows down whilst traveling through mediums other than a vacuum is that the light "takes every possible path at the same time" through the medium. Below I've drawn my two possible interpretations of what that means. Can...
  34. W

    Using Cauchy's integral formula to evaluate integrals

    Homework Statement Use Cauchy’s integral formula to evaluate the integral along γ(t) of (z/(z+9)^2)dz where γ(t) = 2i + 4e^it , 0 ≤ t ≤ 2π. Homework Equations Cauchy's integral formula The Attempt at a Solution I was just wondering is the integral not just zero by Cauchy's theorem since...
  35. O

    How Do You Solve a Second Order Differential Equation Using Trial Solutions?

    Homework Statement (d^2y/dx^2) + (dy/dx) = cos xSo you have the trial solution y= p*cos(x)+q*sin(x) (dy/dx) = -p*sin(x)+q*cos(x) (d^2y/dx^2)=-p*cos(x)-q*sin(x) The issue I am having is equating the coefficients when after I have subbed them into the initial equation...
  36. kapitalpro

    Non-Elementary Integrals: Solving a Challenging Calculus Problem

    Homework Statement Homework Equations The Attempt at a Solution I evaluate the first integral and get (1-cos(x^3))/x then can't go further from that.
  37. N

    Finding work done using definite integrals

    On applying definite integral to find work done, we integrate F.dx and apply lower and upper limits. Should we apply the dot product, before integration , that is -1 for θ = 180, 1 for θ = 0. Or will the limits applied and their values suffice in deciding the sign of the final value. I have...
  38. T

    B Trouble converting definite integrals to Riemann's and back

    can i request anyone to please show me the step by step with specific explanations? thank you! i saw this on stackexchange, and the steps shown are really fuzzy to me :(
  39. egio

    I Difference between integral and antiderivative?

    Hello, I still don't really understand what an antiderivative is, besides its ability to "undo" derivatives, its relation to integrals, and what the difference between the two even is. It would also be great to know how to visualize an antiderivative. I've tried looking further into the...
  40. saybrook1

    Determining Cauchy principal value of divergent integrals

    Homework Statement So I've found a ton of examples that show you how to find cauchy principal values of convergent integrals because it is just equal to the value of that integral and you prove that the semi-circle contribution goes to zero. However, I need to find some Cauchy principal values...
  41. saybrook1

    Still having trouble solving integrals via residues

    Homework Statement Hi guys, I don't quite understand how to solve closed integrals "around a given circle." I was given \oint\frac{dz}{sinz} around |z-6|=4 and said that the integral is equal to zero because the singularity n\pi i is not within the circle. Is this correct? Also I need to...
  42. Alexandra Fabiello

    Concept question - integrals requiring substitution to solve

    Homework Statement Random example: 2[([x^2] + 3)^7](7x) Homework Equations ? The Attempt at a Solution I know that somehow you substitute [x^2] + 3 with 'u', but I don't understand the process going forward for it, and my teacher and textbook has some rather convoluted stuff in it, so if...
  43. JulienB

    I General questions about integrals (definitions)

    Hi everybody! I'm currently studying integrals, and I would like to clarify a few definitions, especially about the criterions of convergence/divergence of an integral. Basically if that's okay for you guys I'm going to list and number a few statements and I'd like to know if they are true or...
  44. S

    MHB Holder inequality for double integrals

    Hi I have a double integral $\iint(f(x) g(x,y)dxdy$ over $x\in[a,y]$ , $y\in[a,b]$ and I wish to bound the integral in terms of integral f times integral g. I suppose there must exist a form of holder inequality to do that ? many thanks Sarrah
  45. S

    Trying to prove trigonometric integrals on a quarter of circle

    Homework Statement I want to prove that: Homework EquationsThe Attempt at a Solution I tried using the trigonometric identity: sen2x = senx cosx / 2, so, I got: 1/2m∫(sen2x)mdx, x from 0 to pi/2, but now I don't know how to proceed. Can you help me please?
  46. matt_crouch

    How do I solve complex contour integrals in complex analysis?

    I am trying to teach myself complex analysis . There seems to be multiple ways of achieving the same thing and I am unsure on which approach to take, I am also struggling to visualise the problem...Would someone show me step by step how to solve for example...
  47. C

    Computation of non scalar loop integrals

    Consider the following integral that comes out of a loop calculation along with some fermionic propagators (e.g virtual one loop correction to a ##p \gamma^* \rightarrow p'## process such as in DIS): $$ \int \frac{\text{d}^d l}{l^2 (l-p)^2 (p+q-l)^2} \text{Tr}(\not p \gamma^{\nu} (\not p + \not...
  48. P

    Integrals and potential energy

    Does the equation http://m.imgur.com/2KAADas Accurately describe the potential energy gained by an object falling by 100,000 units? I asked my physics teacher and he said he didn't know enough about integrals to answer it. I asked my math teacher and she just asked "Are you trying to find the...
  49. heartshapedbox

    Finding future x position with changing velocity

    Homework Statement At what time in the future will the x-component of the masses position be at 0m? Homework Equations x=x,initial+(integral0,t)(v,xcomponent)(dt) The Attempt at a Solution The solution is 3.1 seconds... not really sure where to start :(
  50. anhtu2907

    Proving a theorem in line integrals

    At the bottom of the picture, I couldn't understand why differentiating with respect to x gives the first integral at the right-hand side 0. Thanks for reading.
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