Integration Definition and 1000 Threads

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. 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...
  2. 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...
  3. 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)
  4. 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...
  5. 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
  6. anemone

    MHB Continuous Function Integration Challenge

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

    Integration problem using Integration by Parts

    i would appreciate alternative method...
  8. 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...
  9. karush

    MHB 2.4.3 AP Calculus Exam Integration limits

    by observation I choose (c) since the limit values may not be =
  10. 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...
  11. 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...
  12. 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: " ...
  13. 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: " ...
  14. 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...
  15. 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...
  16. A

    MHB Write on Interpretations of integration (give examples).

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

    MHB Understanding the Meaning of Integration in Math

    what is The meaning of integration?
  18. 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!
  19. E

    B Confused about holding variables constant during integration

    For a double integral, we might treat the "inner integral" separately and be able to compute something like ##\int_{x_1}^{x_2} f(x,y) dx## by holding ##y## constant during the integration. The same technique is applied in other places too, like for solving exact differential equations. I haven't...
  20. LCSphysicist

    I Integrate 1/(x*lnx): Integration by Parts

    can integrate 1/(x*lnx) by parts??
  21. S

    MHB Understanding Example from Topics in Banach Space Integration

    Hey Could you give me a hint how to explain this example? Need help to prove statement in red frame. Example from book (Topics In Banach Space Integration) by Ye Guoju‏، Schwabik StefanThank you
  22. T

    I Surface of a cone by integration

    If i want to calculate the volume of a cone i can integrate infinitesimal disks on the height h of the cone. I was told that if i want to calculate the surface of the cone, this approximation is not correct and i have to take the slanting into account, this means that instead of...
  23. C

    Can indefinite integration be simplified using substitution?

    Let x=t^2 Then dx=2t dt Integral of 1/(x(1-x))^(1/2)dx = integral of 2tdt/t(1-t^2) ^(1/2) = integral of 2dt/(1-t^2) ^(1/2) = 2 arcsin(t) +c = 2 arcsin(rt(x)) +c. But the answer in my book is arcsin(2x-1) +c. Tell me how 2 arcsin(rt(x) +C= arcsin(2x-1) +c I know the constant will vary for both...
  24. M

    I How to Calculate Weights for Gauss-Kronrod Quadrature Using Nodes and Degree?

    The Gauss-Kronrod quadrature uses the zeros of the Legendre Polynomials of degree n and the zeros of the Stieltjes polynomials of degree n+1. These zeros are the nodes for the quadrature. For example using the Gauss polynomial of degree 7, you will need the Stieltjes of degree 8 and both makes...
  25. Sokolov

    Integration of inexact differentials in Thermodynamics

    In Thermodynamics, I have seen that some equations are expressed in terms of inexact differentials, ##\delta##, instead of ##d##. I understand that this concept is introduced to point out that these differential forms are path-dependent, although I am not clear how they can be handled. So, are...
  26. S

    Integration of an exponential and algebra

    ##\int \frac{e^x (2-x^2)}{(1-x) \sqrt{1-x^2}} dx## I tried using substitution x = sin θ but still can not solve it. I guess I have to get rid the term ex but do not know how Thanks
  27. J

    MHB Value of k in definite Integration

    Let p(x)=2x^6+4x^5+3x^4+5x^3+3x^2+4x+2. Let \displaystyle I_{k}=\int^{\infty}_{0}\frac{x^k}{p(x)}dx where 0<k<5. Then value of k for which \displaystyle I_{k} is smallest.
  28. penroseandpaper

    I Integration of a hyperbolic function

    The integral of cothx is ln|sinhx|+C. Does this mean the integral of coth2x is ln|sinh2x|+C? If not, does anyone have a link to a page on how it is achieved - I'm trying to compile a list of all common hyperbolic function derivatives and integrals. However, I can't find anything to confirm if...
  29. A

    Integration in polar coordinates

    In spherical poler coordinates the volume integral over a sphere of radius R of $$\int^R_0\vec \nabla•\frac{\hat r}{r^2}dv=\int_{surface}\frac{\hat r}{r^2}•\vec ds$$ $$=4\pi=4\pi\int_{-\inf}^{inf}\delta(r)dr$$ How can it be extended to get $$\vec \nabla•\frac{\hat r}{r^2}=4\pi\delta^3(r)??$$
  30. Math Amateur

    I Riemann Integration ... Existence Result .... Browder, Theorem 5.12 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...
  31. Math Amateur

    MHB Riemann Integration ... Existence Result .... Browder, Theorem 5.12 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...
  32. 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...
  33. T

    Help with this Integration please

    Summary:: I just need to know how we got from the 'beginning point' to the 'end point'/'answer'. The left side is where we start and my professor did a bunch of calculations so fast that I wasn't able to understand how he got the result on the right side. Could someone help me integrate this...
  34. tworitdash

    A Integration of Bessel's functions

    I can only find a solution to \int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . The closed form solution to \int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho I am not able to find anywhere. Is there any way in which I can approach this problem from scratch...
  35. V

    Finding volume using integration

    I know that the formula for volume is equal to the definite integral ∫A(x)dx, where A(x) is the cross sectional. I found the definite integral where b=5 and a=0, for ∫4x2dx. I obtained the answer 500/3, however this was incorrect, and I'm unsure of where I went wrong? Thank you.
  36. jk22

    I Integration : Are a function and it's derivative independent?

    The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.
  37. M

    Find psi(x,t) when psi(x,0)= Ae^(-x^2/a^2) and A, a are real constants

    EQ 1: Ψ(x,0)= Ae-x2/a2 A. Find Ψ(x,0) So I normalized Ψ(x,0) by squaring the function, set it equal to 1 and getting an A I. A=(2/π)¼ (1/√a) B. To find Ψ(x,t) EQ:2 Ψ(x,t)= 1/(√2π) ∫ ∅(k) ei(kx-ωt)dk --------->when ω=(ħk2)/2m and integral from -∞ to +∞ EQ 3: ∅(k)= 1/(√2π) ∫ Ψ(x,0)...
  38. karush

    MHB 4.2.8 AP calculus Exam Integration limits

    ok I posted a image to avoid any typos but was wondering why the question has dx and options are in dt I picked C from observation but again that was assuming f was a horizontal line of which it could be something else that way the limits stay the same but the area is cut in halfopinions...
  39. jisbon

    Understanding Integration by Substitution

    Not sure how do I start from here, but do I let $$u = lnx$$ and substitute? Cheers
  40. Adesh

    Calculus What are some books for learning the techniques of Calculus?

    We have so many great books available for Calculus, such as : Spivak's Calculus, Stewart Calculus, Thomas Calculus , Gilbert Strang's Calculus, Apostol's Calculus etc. These books are very nice but they teach you the concepts well and all the standard techniques that are available for solving...
  41. C

    A change in the order of integration

  42. Physics lover

    Challenging Integral Homework: Attempting x=tanA/b Substitution

    Homework Statement: The question is in Attempt at a solution. Homework Equations: x=tanA/b I tried by substituting x=tanA/b but it did'nt helped.Now I cannot think of any other thing to do.Help.
  43. anita chandra

    A Does this integration have a closed form solution?

    I was trying to solve a differential equation that I defined to study the dynamics of a system. Meanwhile, I encounter integration. The integration is shown in the image below. I tried some solutions but I am failed to get a solution. In one solution, I took "x" common from the denominator terms...
  44. D

    Calculating Divergent Amplitude in Phi-4 Theory

    For the diagram In scalar field theory, I have obtained an integral which looks like $$\int_{0}^{\Lambda} \frac{d^4 q}{(2\pi)^4} \frac{i}{q^2 - m^2 + i\varepsilon} \frac{i}{(p - q)^2 - m^2 + i\varepsilon}$$ I am required to calculate this and obtain the divergent amplitude $$i\mathcal{M} =...
  45. Z

    Integration limitations to a SiC microprocessor

    There has been a demonstration of SiC BJT ADCs, and SiC JFET SRAM. What would be the major limitations associated with higher forms of integration for SiC? For example, a 1billion transistor CPU based on SiC?
  46. E

    Definite and indefinite integration in the definition of work

    This is going to sound like a silly question, but here we go anyway! I've always thought about a definite integral being used for modelling a change in some quantity whilst an indefinite integral is employed to find the defining function of that quantity. For example, consider the...
  47. George Keeling

    I Is There a Better Way to Solve Integrals Than Symbolab and Mathematica?

    I remember being given a ghastly book of integrals to learn when I was about 16. I went to sleep. Apparently the first book of integrals was published by Meier Hirsch in 1810. There have been many more since then. Surely with the invention of the internet there is something better? Symbolab has...
  48. B

    I Integration: When to multiply by one or add zero?

    I have seen several functions be integrated by multiplying by a form of one or by adding a form of zero. When is it advantageous do do one of these things? Are there any example problems (Calc I or II) in which I can try these techniques?
  49. G

    How to use the Double integration method for an overhanging beam?

    In case of overhanging beam with point load at the end. For example: (here RA-reaction is negative) The equation will be as follows (by double integration method): , as we can see the equation will not have Point load (10kN) term in it. 1) How the influence of the point load is accounted in...
  50. mastrofoffi

    I Integration of Poisson's Equation

    I have a gaussian charge distribution, in gaussian units $$ \rho(\mathbf r) = q\frac{\alpha^3}{\pi^{3/2}}\exp( -\alpha^2 r^2 ) $$ and I want to solve Poisson's equation to find the electrostatic potential $$ \nabla^2 \psi(\mathbf r) = -4\pi\rho(\mathbf r). $$ Since the charge distribution has...
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