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. U

    I Unclear step in "Change of variable in a multiple integral" proof

    I'm studying the proof of this theorem (Zorich, Mathematical Analysis II, 1st ed., pag.136): which as the main idea uses the fact that a diffeomorphism between two open sets can always be locally decomposed in a composition of elementary ones. As a remark, an elementary diffeomorphism...
  2. T

    I How can you prove the integral without knowing the derivative?

    Hello (A continued best wishes to all, in these challenging times and a repeated 'thank you' for this site.) OK, I have read that Newton figured out that differentiation and integration are opposites of each other. (This is not the core of my question, so if that is wrong, please let it go.)...
  3. MathematicalPhysicist

    Change of variables in a simple integral

    So we have ##x=\beta(1/2 mv^2-\mu)##, i.e ##\sqrt{2(x/\beta+\mu)/m}=v##. ##dv= \sqrt{2/m}dx/\sqrt{2(x/\beta+\mu)/m}##. So should I get in the second integral ##(x+\beta \mu)^{1/2}##, since we have: $$v^2 dv = (2(x/\beta+\mu)/m)\sqrt{2/m} dx/\sqrt{2(x/\beta+\mu)/m}$$ So shouldn't it be a power...
  4. D

    Analyzing a Complex Line Integral Using Substitution and Logarithmic Properties

    if ## \gamma (t):= i+3e^{2it } , t \in \left[0,4\pi \right] , then \int_0^{4\pi} \frac {dz} {z} ## in order to solve such integral i substitute z with ##\gamma(t)## and i multiply by ##\gamma'(t)## that is: ##\int_0^{4 \pi} \frac {6e^{2it}}{i+3e^{2it}}dt=\left.log(i+3e^{2it}) \right|_0^{4...
  5. anemone

    MHB Integral of trigonometric function

    Prove that if $[a,\,b]\subset \left(0,\,\dfrac{\pi}{2}\right)$, $\displaystyle \int_a^b \sin x\,dx>\sqrt{b^2+1}-\sqrt{1^2+1}$.
  6. anemone

    MHB Integral Challenge: Evaluating $\int_0^\infty \frac{x^2+2}{x^6+1} \, dx$

    Evaluate $\displaystyle\int\limits_0^{\infty} \dfrac{x^2+2}{x^6+1} \, dx$.
  7. hyksos

    A Derive the Principle of Least Action from the Path Integral?

    Several weeks ago I had considered the question as to how one can start from the Schroedinger Equation, and after several transformations, derive F=ma as a limiting case. At some point in my investigations of this derivation, it occurred to me that this is simply too much work. While in...
  8. D

    Using a Surface Integral for Mathematical Analysis of the Area of an Island

    I am not clearly understand what the question requests for, is it okay to continue doing like this ? Kindly advise, thanks
  9. D

    Evaluate integral using Green Theorem

    I got stuck here, how to integrate e^(y^2), I searched but it's something like error function
  10. D

    Is the Triple Integral in Cylindrical Coordinates Correctly Solved?

    I am trying to solve it using cylindrical coordinates, but I am not sure whether the my description of region E is correct, whether is the value of r is 2 to 4, or have to evaluate the volume 2 times ( r from 0 to 4 minus r from 0 to 2), and whether is okay to take z from r^2/2 to 8
  11. A

    A How can we approximate the following integral for large D?

    How to solve the following integral (in Maple notation): Int(y**k*exp(-u[0]*exp(-y)/a[0]-u[1]*exp(y)/a[1]),y=-infinity..infinity) with 0<a[0], 0<u[0], 0<a[1], 0<u[1]?
  12. A

    A How to solve this integral? (something to do with a beta distribution?)

    I have the following integral (in Maple notation): Int(exp(c[0]*ln(y)/a[0]+c[1]*M*ln(M-y)/a[1]), y = 0 .. M); with (in Maple notation): 0<a[0], 0<a[1], 0<c[0], 0<c[1], 0<y, y<M, 0<M. What is the solution of this integral? I suspect that the solution has something to do with a beta distribution.
  13. S

    B Solving for integral curves- how to account for changing charts?

    [Ref. 'Core Principles of Special and General Relativity by Luscombe] Let ##\gamma:\mathbb{R}\supset I\to M## be a curve that we'll parameterize using ##t##, i.e. ##\gamma(t)\in M##. It's stated that: Immediately after there's an example: if ##X=x\partial_x+y\partial_y##, then ##dx/dt=x## and...
  14. agnimusayoti

    Indefinite integral of cross product of 2 function

    I've tried with this work in attachment. i&m not sure of my answer is correct.
  15. A

    I Derivative of a definite integral

    If $$F(x)=\int_{a}^{b}f(x)dx$$ implies $$F'(x)=\int_{a}^{b}f'(x)dx$$?
  16. agnimusayoti

    Exploring Multi-Variable Integral Limits with $\sin(t)/t$

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  17. O

    MHB Can Improper Integrals Help Solve This Inequality?

    This is my method, could you help me to continue?
  18. Adesh

    I Understanding the ##\epsilon## definition of this integral

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  19. H

    I Estimate the magnitude of a line integral exp(iz) over a semicircle

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  20. R

    How to solve the integral which has limits from (1,2) to (2,4)

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  21. T

    Multivariable Calculus, Line Integral

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  22. Q

    Solving Tricky Integral: How to Proceed Further?

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  23. HexHammer

    B What Is the Occurrence Rate of Path Integral in a Photon Double-Slit Experiment?

    Hi forgive my very ignorant question. How frequently does 'Path Integral', curved shot and normal shot happen out of say 100 shots with a photon?
  24. tworitdash

    A Integral of a sinc squared function over a square root function

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  25. karush

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  26. tworitdash

    A Spectral domain double integral with singularities

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  27. L

    MHB Integral limits when using distribution function technique

    I am not sure about finding the limit of the integral when it comes to finding the CDF using the distribution function technique. I know that support of y is 0 ≤y<4, and it is not a one-to-one transformation. Now, I am confused with part b), finding the limits when calculating the cdf of Y...
  28. thaiqi

    I Variation sign and integral sign

    Hello, everyone. I know that it is feasible to exchange the order of one variation sign and one integral sign. But there gives a proof of this in one book. I wonder about a step in it. As below marked in the red rectangle: How can ##\delta y## and ##\delta y^\prime## be moved into the integral...
  29. J

    Reducing Bessel Function Integral

    I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing...
  30. Saracen Rue

    I Find the value of this definite integral in terms of t, s and alpha

    One of the maths groups I'm apart of on Facebook posts (usually) daily maths challenges. Typically they act as small brain teaser for when I wake up and I can solve them without much trouble. However, today's was more challenging: (Note: blue indicates a variable and red indicates a constant)...
  31. tworitdash

    A Spatial Fourier Transform: Bessel x Sinusoidal

    I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...
  32. S

    I Help with a 3D Line Integral Problem (segmented line)

    Hi all, I'm finding it difficult to start this line integral problem. I have watched a lot of videos regarding line integrals but none have 3 line segments in 3D. If someone can please point me in the right direction, it would help a lot. I've put down the following in my workings: C1...
  33. G

    How do I solve for the centroid of a function with a given range?

    Summary:: I'm solving an exercise. I have the following center of gravity problem: Having the function Y(x)=96,4*x(100-x) cm, where X is the horizontal axis and Y is the vertical axis, ranged between the interval (0, 93,7) cm. Determine: a) Area bounded by this function, axis X and the line...
  34. G

    Line integral where a vector field is given in cylindrical coordinates

    What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
  35. Physics lover

    No. of positive integral solutions of fractional functions

    I know how to find integral solutions of linear equations like x+y=C or x+y+z=C where C is a constant. But I don't have any idea how to solve these type of questions.I am only able to predict that both x and y will be greater than 243554.Please help.
  36. dykuma

    Evaluating an integral of an exponential function

    the integral is: and according to mathematica, it should evaluate to be: . So it looks like some sort of Gaussian integral, but I'm not sure how to get there. I tried turning the cos function into an exponential as well: however, I don't think this helps the issue much.
  37. jaumzaum

    I Possibility of an integral system

    I was solving a Physics problem, and for it to be consistent there should exist a function f(t) in real numbers and a time T, such that: $$\int_{0}^{T} f(t) dt=0 $$ $$\int_{0}^{T} \int_{0}^{t} f(t') dt' dt=0$$ $$\int_{0}^{T} f(t) (\int_{0}^{t} f(t') dt') dt>0$$ i.e. the integral is zero, the...
  38. Replusz

    I Invert an integral equation to find this unknown

  39. Replusz

    I Changing to Polar coordinates in order to calculate this integral

    m a
  40. B

    A Can this difficult Gaussian integral be done analytically?

    Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...
  41. M

    MHB Calculate the integral using the Fourier coefficients

    Hey! :o A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$. I want to calculate $\int_0^{T_0}x^2(t)\, dt$. I have done the following: It holds that...
  42. P

    MHB Solving Integral Equation w/ Laplace Transform - Abdullah

    We would need to recognise that the integral in the equation is a convolution integral, which has Laplace Transform: $\displaystyle \mathcal{L}\,\left\{ \int_0^t{ f\left( u \right) \,g\left( t - u \right) \,\mathrm{d}u } \right\} = F\left( s \right) \,G\left( s \right) $. In this case...
  43. C

    A Evaluation of an improper integral leading to a delta function

    Hi, I have pasted two improper integrals. The text has evaluated these integrals and come up with answers. I wanted to know how these integrals have been evaluated and what is the process to do so. Integral 1 Now the 1st integral is again integrated Now the text accompanying the integration...
  44. O

    I Am I using the right limits on this triple integral?

    Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} Consider the expression: \begin{align} \int_{q'}^q \int_{b'}^b g(s)\ \int_{s-t}^{s+t} f(r)\ dr\ ds\ dt\ \end{align} We...
  45. J

    Confused with this integral from Griffith's electrodynamics.

  46. Avatrin

    Parametric distance of a line in a grid (Line Integral Convolution)

    Hi, the above image is from the Line Integral Convolution paper by Cabral and Leedom. However, I am having a hard time implementing it, and I am quite certain I am misreading it. It is supposed to give me the distances of the lines like in the example below, but I am not sure how it can. First...
  47. F

    Solving the Integral ∫dx/(1-x)

    I solved the integral by two different methods and I get different answers. Method 1: ∫dx/(1-x) = -∫-dx/(1-x), u=1-x, du=-dx ∫dx/(1-x) = -∫du/u = -ln|u| = -ln|1-x| Method 2: ∫-dx/(x-1) = -∫dx/(x-1), u=x-1, du=dx ∫-dx/(x-1) = -∫du/u = -ln|u| = -ln|x-1| What am I doing wrong?
  48. G

    Integral Involving the Dirac Delta Function

  49. benorin

    Insights A Path to Fractional Integral Representations of Some Special Functions

    Introduction This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets $$C_N^n: = \left\{ { \vec x \in {\mathbb{R}^n}|{x_i} \ge 0\forall i,\sum\limits_{k = 1}^n {x_k^{2N}} < n - 1 } \right\}$$ and Dirichlet integrals to...
  50. C

    A Square of an integral containing a Green's Function

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