What is Integration: Definition and 1000 Discussions

System integration is defined in engineering as the process of bringing together the component sub-systems into one system (an aggregation of subsystems cooperating so that the system is able to deliver the overarching functionality) and ensuring that the subsystems function together as a system, and in information technology as the process of linking together different computing systems and software applications physically or functionally, to act as a coordinated whole.
The system integrator integrates discrete systems utilizing a variety of techniques such as computer networking, enterprise application integration, business process management or manual programming.System integration involves integrating existing, often disparate systems in such a way "that focuses on increasing value to the customer" (e.g., improved product quality and performance) while at the same time providing value to the company (e.g., reducing operational costs and improving response time). In the modern world connected by Internet, the role of system integration engineers is important: more and more systems are designed to connect, both within the system under construction and to systems that are already deployed.

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  1. E

    I Integration trouble (integral over a 2-sphere)

    There's an integral over a 2-sphere ##S## with unit normal ##N^a## within a hypersurface orthogonal to a Killing field ##\xi^a##$$F = \int_S N^b (\xi^a / V) \nabla_a \xi_b dA = \frac{1}{2} \int_S N^{ab} \nabla_a \xi_b dA, \quad N^{ab} := 2V^{-1} \xi^{[a} N^{b]}$$which follows because the Killing...
  2. tanaygupta2000

    Integration of coherent state

    I began this solution by assuming a = x+iy since a is a complex number. So I wrote expressions of <a| and |a> in which |n><n| = I. I got the following integral: Σ 1/πn! ∫∫ dx dy exp[-(x^2 + y^2)] (x^2 + y^2)^n I I tried solving it using Integration by Parts but got stuck in the (x^2 + y^2)^n...
  3. R

    Integration of abs(k)e^(ikx)dk

    Split the integral $$\frac{Aa}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{ikx}dk - \frac{A}{\sqrt{2\pi}}\int^{\infty}_{-\infty}|k|e^{ikx}dk$$ Apply the boundary conditions, this is where my biggest source of uncertainty comes from I doubled the integral and integrated from 0 to a instead of from -a...
  4. Santiago24

    Calculus Calculus textbooks with good sections on integration

    Hi I'm having troubles with integration specially by substitution, I'm going to read a calculus textbook and i need recommendations of books with a good treatment on the different techniques of integration. I'd like a book with good exercises for self study and a exposure to integration of...
  5. S

    Integration of this trigonometry function

    Is it possible to do the integration? That is the full question I don't know where to start, try to use ##u=\cos x## and also ##\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)## but failed. Thanks
  6. A

    Problem with setting the region of integration

    Good day ! I have a problem with the solution of the floowing integrals Indeed i don't understand why they choose such borders for integral b/a<c y<c doesn't mean that y<b/a ! many thanks in advance!
  7. Mayhem

    B Why don't we account for the constant in integration by parts?

    As we all know, integration by parts can be defined as follows: $$\int u dv = uv - \int v du$$ And the usual strategy for solving problems of these types is to intelligently define ##u## and ##dv## such that the RHS integral can easily be evaluated. However, something that is never addressed is...
  8. JD_PM

    Integration and hyperbolic function problem

    This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...
  9. S

    Understanding Griffith's Velocity Argument for Charge Integration

    In Griffith’s section 10.3.1, when proving why there is an extra factor in integrating over the charge density when it depends on the retarded time, he makes the argument that there can only ever be one point along the trajectory of the particle that “communicates” with the field point. Because...
  10. S

    B Integration of tan^2 x from - to + infinity

    ∫tan^2 x ( -infinity to +infinity)
  11. D

    I Help With a Proof using Contour Integration

    I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof. He is evaluating an integral along a contour which makes up the imaginary axis. He has the following: $$ -i\int_{-iR}^{iR}...
  12. S

    B Integration from "Area Under Curve" Perspective: Explained

    I can calculate the value of the integration, it will be ##\frac{\sqrt{3}}{2}## But if I draw the function and consider the area bounded by the curve and x-axis from x = 0.5 to x = 1, it seems that the area will be infinite because x = 1 is vertical asymptote. Why can't I consider from "area...
  13. M

    Integrating Partial Fractions with a Twist: (x + 3) vs (x - 3)

  14. chwala

    Integration of a trig function

    This is my first attempt ...
  15. chwala

    Integration of a trigonometic function

    my thinking was to have everything changed to a function that has cosine only... ##\int_0^{0.5π} \frac {1-cos^2x}{sin x + cos x}dx## ##\int_0^{0.5π} \frac {(1-cos x)(1+cos x)}{(1-cos^2x)^{0.5} + cos x}dx## ... first of all is this integration possible? if so then let me know if i am on the...
  16. Tony Hau

    I How to interpret integration by parts

    So I am confused about a proof in which the formula for expected value of velocity, ##\frac{d\langle x \rangle}{dt} ##, is derived. Firstly, because the expected value of the position of wave function is $$\langle x \rangle =\int_{-\infty}^{+\infty} x|\Psi(x,t)|^2 dx$$Therefore...
  17. R

    I Is it possible to solve such a differential equation?

    Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...
  18. WannaLearnPhysics

    Using Ampere's Law for these two different integration paths

    Homework Statement:: The magnetic field at every point on the path of integration Relevant Equations:: The scenarios/situations are shown in the attached photo. "Any conductors present that are not enclosed by a particular path may still contribute to the value of B field at every point, but...
  19. L

    Calculus and Kinematic equations--- seeing the logic

    Details of Question: ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into: s − s0 = v0t + ½at2 My main question is about the integration of...
  20. J

    I Integration of differential forms

    I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example: Let ##M## be a smooth...
  21. N

    Integration by parts on ##S^3## in Coleman's textbook

    I'm reading Coleman's "Aspects of symmetry" chap 7. On the topic of the SU(2) winding number on ##S^3##on page 288, three parameters on ##S^3## are defined ##\theta_1,\theta_2,\theta_3##. Afterwards, it defines the winding number and to show it's invariant under continuous deformation of gauge...
  22. P

    A What is the meaning of ##d\Omega## in solid angle integration?

    Anyone have any idea how to perform the following two integrals? ##\int d\Omega n_{i}n_{j}## and ##\int d\Omega n_{i}n_{j}n_{k}n_{l}## where the n is a unit vector.
  23. B

    Integrating with a Denominator of (1+x^2): A Step-by-Step Guide

    I think in the case of "n da" you can see the denominator (1+x^2) as a constant, so ∫ ( sin(a) + M^2 ) / ( 1 + x^2 ) da = ( 1 / ( 1 + x^2 ) ) * ∫ (sin(a) + M^2 ) da = ( 1 / ( 1 + x^2 ) ) * ( -cos(a) + (M^2)a ) = ( - cos(a) + (M^2)a ) / ( 1 + x^2 ) --- Is this the way to go? This is my...
  24. T

    I Bose-Einstein numerical integration

    Want to integrate the total energy density over all photon energies between two temperature values from 500K to 5800K, but not sure how to proceed. Here is some examples to help:
  25. jaychay

    MHB Can Integration by Parts Solve This Tricky Question?

    Can you help me with this question ? I am really struck with this question. Thank you in advance.
  26. jaychay

    MHB Can You Help Me Integrate $\sin(u) + u^6$ Correctly?

    Can you please help me ? I have tried to do it many times but still got the wrong answer. Thank you in advance.
  27. H

    Integration ## f(\theta, \phi) = \frac{sin \theta}{4\pi}##

    Hi, I have this formula ## f(\theta, \phi) = \frac{sin \theta}{4\pi}## I have this statement that say if I integrate this formula above on a sphere then p = 1. what does integrate on a sphere means? I know ##\int_0^{2\pi} ## is used for the circle.
  28. BWV

    A Integration with respect to a Lévy basis / Ambit fields

    Familiar with basics of stochastic calculus and integration over a Brownian motion. Trying to get a sense of Ambit Fields https://en.wikipedia.org/wiki/Ambit_field which mention an integration over a Lévy basis: Curious if anyone familiar with this? A Brownian motion is a Levy process...
  29. greg_rack

    B How to learn differentiation and integration in 14 days?

    The detailed list of the concepts I should master I'm attending the last year of high school and I'm currently studying limits. For university test reasons I'll need to study on my own topics such as differentiation and integration... and I have just 14 days to do so! Firstly, do you think it's...
  30. D

    Reversing the order of integration in a double integral

    Performing the x-integration first the limit are x=y2 and x= -y2 and then the y limits are 0 to 1. This gives the final answer 2/5 But i am getting confused when trying to reverse the order of integration. My attempt is that i have to divide the region in 2 equal halfs and then double my answer...
  31. H

    Integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}##

    Hi, I'm trying to integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}## and ## \int \frac{pdz \cdot \hat p}{(\sqrt{p^2 + z¸^2)^3}}## I get ## \frac{\hat z}{(\sqrt{p^2 + z¸^2)}}## and ## \frac{2z \cdot \hat p}{(\sqrt{p^2 + z¸^2)}}## But the correct answer should be ## \frac{z \hat...
  32. H

    Integration ##\ddot\phi = -\omega^2\phi##

    Hi, I'm wondering how can I get ## \phi(t) = A sin(\omega t) + B cos(\omega t)## I know I have to integrate 2 times ##\ddot\phi = -\omega^2\phi##. However, I don't have any more explanation in my book. I know A and B are the constants of integration.
  33. Remixex

    Contour integration around a complex pole

    $$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$ So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise. The generalized complex integral would be $$\int_{C} \frac{e^{-i \alpha...
  34. T

    A Integration of the Maxwell Speed Distribution

    Need some help on how to solve the integration formula for Maxwell speed distribution, here is the procedure on how to solve for the kinetic energy: Not familiar with the error function yet, but the result for the kinetic energy integration is...
  35. Hamiltonian

    Calculus Books for practice integration problems and calc 3

    I am looking for a (practice)book that has problems on definite and indefinite integration from easy to intermediate. also which book covers the prerequisites of calculus for books like Griffiths.(similar to the topics in chap 1 of Griffiths but more in-depth)
  36. A

    Problem figuring out the surface of integration

    Good day I have a problem figuring out the surface of integration according to the exercice, we have a paraboloid that cross a disk on the xz plane, the parabloid cross the xz plane on a smaller disk r=√3/3 so for me after going to the final step of integration and using polar coordinate i...
  37. A

    Different results while switching the order of integration

    my Problem is that I get a different result when I switch the order of integration (X over Y), I couldn't spot the mistake, any help would highlyu appreciated
  38. anemone

    MHB Continuous Function Integration Challenge

    Find all continuous functions $f:[1,\,8] \rightarrow \mathbb{R} $ such that $\displaystyle \int_1^2 f^2(t^3)dt + 2\int_1^2 f(t^3)dt=\dfrac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt$
  39. chwala

    Integration problem using Integration by Parts

    i would appreciate alternative method...
  40. rude man

    I Confusion on a simple integration

    The tables and Wolfram Alpha say ## \int dx/(x^2 + a^2) ^{1/2} = log~ [( x^2 + a^2)^{1/2} + x] ##. So if a=0 we get as answer ## log (x + x) = log( 2x) ## if ## (x^2)^{1/2} = x, or ## log (x - x) = log( 0) ## if ## (x^2)^{1/2} = -x but surely ## \int dx/(x^2)^0.5 = \int...
  41. karush

    MHB 2.4.3 AP Calculus Exam Integration limits

    by observation I choose (c) since the limit values may not be =
  42. A

    I Converting a summation into an integration

    Hello, I want to convert a summation in reciprocal space and I am unsure about the integration volume. I have started with the formula: $$\sum_{\vec{k}} \rightarrow \frac{V_{k}}{(2\pi)^{3}}\int\int\int \mathrm{d}V_{k}$$ where: $$\mathrm{d}V_{k} = k^{2}\mathrm{d}k...
  43. agnimusayoti

    Changing Variables and the Limits of Integration using the Jacobian

    From the equations, I can find Jacobians: $$J = \frac {1}{4(x^2 + y^2)} $$ But, I confuse with the limit of integration. How can I change it to u,v variables? Thanks...
  44. Math Amateur

    MHB Lebesgue Integration of Simple Functions .... Lindstrom, Lemma 7.4.6 .... ....

    I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...
  45. Math Amateur

    I Lebesgue Integration of Simple Functions .... Lindstrom, Lemma 7.4.6 ...

    I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...
  46. nughii

    Error in trapezoidal integration using a Programming language

    Summary:: I want to iterate a mathematical model using a programming language. The equation of the mathematical model is simple. The following is a brief explanation. I want to iterate a mathematical model using a programming language. The equation of the mathematical model is simple. The...
  47. J

    Need help deducing the region for this double integration problem

    Converting to a polar integral : Integrate ##\(f(x, y)=\) \(\left[\ln \left(x^{2}+y^{2}\right)\right] / \sqrt{x^{2}+y^{2}}\)## over the region ##\(1 \leq x^{2}+y^{2} \leq e\)## So, \begin{array}{c} 1 \leq x^{2}+y^{2} \leq e \\ 1 \leq x^{2} \leq e \quad 1 \leq y^{2} \leq e \\ 1 \leq x \leq...
  48. A

    MHB Write on Interpretations of integration (give examples).

    write on Interpretations of integration (give examples).
  49. A

    MHB Understanding the Meaning of Integration in Math

    what is The meaning of integration?
  50. minimoocha

    MHB Solve Complex Integration: Find 2.36 Area of y=-x/2e+1/e+e & y=e^x/2

    The area of two lines that I need to find is 2.36, however i need this in exact form. The lines are y=-x/2e+1/e+e the other line is y=e^x/2 Since y=-x/2e+1/e+e is on top it is the first function. A=(the lower boundary is 0 and the top is 2) -x/2e+1/e+e-e^x/2 If you could please help!
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