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. U

    Complex Integration Along Given Path

    From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
  2. F

    Jacobian: how to change limits of integration?

    Hello, I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##. From there, I can reformulate with respect to ##z## and...
  3. T

    Integration of acceleration in polar coordinates

    I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image. My attempts are the following, I proceed using 3 "independent" methods just as you...
  4. A

    I Integration of √(1-x^2) with x=sinθ: Check Correctness and Simplification

    Does my below integration is correct? ##\int \sqrt{1 - x^2} \ dx## Let ##x = \sin \theta##, then ##dx = \cos \theta \ d \theta##, ##\cos \theta = \sqrt{1 - x^2}##, ##\theta = \sin^{-1} (x)## ##\int \sqrt{1 - x^2} \ dx## ##= \int \left( \sqrt{1 - \sin^2 \theta} \right) \ \left( \cos \theta \...
  5. A

    I Formula for integration of natural coordinates over an element

    In a textbook I own a formula is given for the integration of natural coordinates over an element. In this case it is a 1 dimensional element (i.e. a line segment) with coordinates ##x_i## and ##x_j##. The coordinate ##x## over the element is written as: $$ x = L_1(x) x_i + L_2(x) x_j $$ with...
  6. benorin

    I I would like opinions of the latest draft of my note - Integration

    The note is entitled: Evaluation of a Class of n-fold Integrals by Means of Hadamard Fractional Integration. 4 pgs pdf format. I assure you that you need not know anything about fractional calculus at all to understand this note that Howard Cohl helped me with. We only use a single...
  7. brotherbobby

    Finding Area of Shaded Segment in Circle Using Calculus

    Problem Statement : To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP. Attempt 1 (without calculus): I mark some relevant lengths inside the circle, shown left. Clearly OS = 9 cm and SP = 12 cm using the Pythagorean...
  8. H

    Proving that ##T## is skew-symmetric, inner product is an integration.

    ##\langle T(f), g \rangle = \int_{0}^{1} \int_{0}^{x} f(t) dt ~ g(t) dt## As ##\int_{0}^{x} f(t) dt## will be a function in ##x##, therefore a constant w.r.t. ##dt##, we have ##\langle T(f), g \rangle = \int_{0}^{x} f(t) dt ~ \int_{0}^{1} g(t) dt## ##\langle f, T(g)\rangle = \int_{0}^{1} f(t)...
  9. Bertin

    Applied Bibliography on integration and ODE/PDE solving techniques for physics

    Hi, you all, I open this thread to ask for any recommendation concerning integration as well as ODE/PDE solving techniques for physics. I love mathematics, and I usually read material on pure mathematics (most notably abstract algebra and a bit of topology) but here I'm more interested in the...
  10. WMDhamnekar

    Change in the order of integration in triple integrals

    If we solve the L.H.S. of this equation, we get ## \frac{(b-a)^3}{6}## and if we solve R.H.S. of this equation, we get ##-\frac{2b^3-3ba^2 +a^3}{6}## So, how can we say, this equation is valid? By the way, how can we use the hint given by the author here?
  11. Samama Fahim

    Limits of Integration in the Transmission Coefficient

    Initially '0' is the upper limit and ##a = \frac{Ze^2}{E}## is the lower limit. With change of variable ##x = \frac{Er}{Ze^2}##, for ##r=0##, ##x=0##, and for ##r=\frac{Ze^2}{E}##, ##x=1##, so 1 should be the lower limit. However, he takes 1 as the upper limit, and without a minus sign. Why is...
  12. A

    A Feynman parametrization integration by parts

    How can i move from this expression: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+i(k-k_{f}))^3} \frac{1}{(1+i(k-k_{i}))^3}$$ to this one: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+|k-k_{i}|^2)^2} \frac{1}{(1+|k-k_{f}|^2)^2}$$ using Feynman parametrization (Integration by...
  13. fresh_42

    Insights The Amazing Relationship Between Integration And Euler’s Number

    Continue reading...
  14. M

    Indefinite Integration of Heaviside function muliplied by a function

    Will it be [{(r-a)/r}*H(r-a)]
  15. M

    Mathematica Numerical integration over a Green's function

    Hi PF! I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this NIntegrate[-(1/((-1.` + x)^2 (1.` + x)^2 (1.` + y)^2)) 3.9787262092516675`*^14 (3.9999999999999907` + x (-14.99999999999903` + x (20.00000000000097` -...
  16. jedishrfu

    I High schooler Develops new Integration Technique

    https://www.wuft.org/news/2022/02/18/buchholz-high-school-student-discovers-and-publishes-new-calculus-technique/
  17. Math Amateur

    MHB Exploring Proposition 6.1.2 from D&K's Multidimensional Real Analysis II (Integration)

    I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ... I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of...
  18. runningphysics

    Solving Motion Equations with Integration

    I'm not sure where to start, when I tired using integration of the initial equation to get pos(t)=-.65t^2 i + .13t^2 j + 14ti +13tj but after separating each component, i and j, and setting j equal to zero I got 0 or -100 seconds which doesn't seem like a reasonable answer.
  19. rudransh verma

    Integration of velocity to get displacement

    Integration of v= integration of##(alpha \sqrt x)dx##. But I am getting wrong answer.
  20. L

    I Number of integration constants

    If we have system of 3 ordinary differential equation in mechanics and we have two initial condition ##\vec{r}(t=0)=0## and ##\vec{v}(t=0)=\vec{v}_0 \vec{i}##. If we somehow get \frac{d^2v_x}{dt^2}=-\omega^2v_x then v_x(t)=A\sin(\omega t)+B\cos(\omega t) Two integration constants and one initial...
  21. S

    Integration of a function

    I tried using substitution ##u=\sqrt{16x-x^8}##, didn't work Tried factorize ##x## from the denominator and then used ##u=\sqrt{16-x^7}##, didn't work Tried using ##u=x^4## also didn't work How to approach this question? Thanks
  22. P

    Notation clarification: SU(N) group integration

    Hello, I would like help to clarify what det( {\delta \over \delta J}) W(J) (equation 15.79) actually means, and why it returns a number (and not a matrix). This comes from the following problem statement (Kaku, Quantum Field Theory, a Modern Introduction) Naively, one would define det...
  23. W

    MHB Double integration problem

    does anyone know how to solve this/can lead me on a direction to where I will get an answer that actually makes sense lol? I keep getting a negative answer/0. For context, I'm tryna find the surface area of a pringle and all the sources I've visited always estimate the projected 2D region as a...
  24. T

    I Impulsive force and simple integration

    May I ask if the following process is correct? Given: F=ma Apply an impulsive force using the dirac delta near 0 (with F nearly constant over the tiny impulsive interval) ma = Fδ(t) This is a second order differential equation with a forcing function. However, I cannot readily integrate...
  25. A

    Changing the interval of integration

    Greetings Dear community! Here is the solutions using two different methods: the first method is the Green theorem and the second is the simple path integration method: My question is why they integrate over [0.2pi] in the path integration method while they integrate within [0. pi] in the...
  26. J

    A Problem solving forces with pressure integration

    Angle theta is different for every place at airfoil surface, so it can't be one theta from leading edge to trailing edge. Can please someone explain pressure integration in depth, step by step?
  27. LCSphysicist

    Complex integration is giving the wrong answer by a factor of two

    $$\int_{0}^{2\pi } (1+2cost)^{n}cos(nt) dt$$ $$e^{it} = z, izdt = dz$$ $$\oint (1+e^{it}+e^{-it})^{n}\frac{e^{nit}+e^{-nit}}{2} \frac{dz}{iz} = \oint (1+z+z^{-1})^{n}\frac{z^{n}+z^{-n}}{2} \frac{dz}{iz}$$ $$\oint (z+z^{2}+1)^{n}\frac{z^{2n}+1}{z^{2n+1}} \frac{dz}{2i} = \pi Res = \pi...
  28. C

    Four dimensional integration in cpp using gsl

    I am trying to perform a four dimensional integration in cpp using gsl. I do this by nesting together the one-dimensional integration routines from gsl library. What I wrote seems to work for a few test integrands but I am having trouble with the integrand that I actually want a value for. See...
  29. A

    Integration of (e[SUP]-√x[/SUP])/√x

    (e-√x)/√x (integral from title) I integrated by substituting and the bounds changed with inf changing to -inf and 1 changing to -1 My final integrated answer is -2lim[e-√x]. What happens to this equation at -inf and -1? As I can't put them into the roots
  30. uzi kiko

    Python Numerical integration over a disk with polar coordinates

    In my job, I was given the task of calculating a force that operates an ultrasound transmitter on a receiver. The calculation is made by assuming that each point on the transmitter is a small transmitter and integration should be made on the surface of the transmitter. Since the transmitter is...
  31. Z

    Calculate the electric field due to a charged disk (how to do the integration?)

    I am interested in particular in the second integral, in the ##\hat{r}## direction. Here is my depiction of the problem: As far as I can tell, due to the symmetry of the problem, this integral should be zero. $$\int_0^R \frac{r^2}{(x^2+r^2)^{3/2}}dr\hat{r}$$ I don't believe I need to...
  32. Rikudo

    Integration in angular momentum

    https://www.physicsforums.com/threa...f-a-translating-and-rotating-pancake.1005990/ So,I think I posted this in the wrong place. So, I will move it to here. Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant. Hence, isn't...
  33. A

    Calculus Textbook for Integration using Hyperbolic substitution

    Can someone please tell me the book that contain integration using hyperbolic substitution for beginner? I know that hyperbolic functions is taught in Calculus book but most of them is only some identities and inverses of hyperbolic functions.
  34. ergospherical

    I PF Integral Bee: Share Interesting/Quirky Integrals!

    Thought it could be fun to have a sort of "PF Integral Bee"... if you know some interesting/quirky/etc. integrals then post them here! 🤓 To get the ball rolling... 1. ##\displaystyle{\int_0^1} \dfrac{\ln{(x+1)}}{x^2+1} dx##
  35. A

    I Integration Using Hyperbolic Substitution

    Can someone please show me an example of integration using hyperbolic substitution? Thank you
  36. Istiak

    How integral and gradient cancels?

    I know that gradient is multi-variable derivatives. But, here line integration (one dimensional integral) had canceled gradient. How?
  37. Istiak

    How to find the constant in this indefinite integration?

    $$x(t)=\int \dot{x}(t)\mathrm dt=vt+c$$ That's what I did. But, book says $$x(t)=\int \dot{x}(t)\mathrm dt=x_0+v_0 t+ \frac{F_0}{2m}t^2$$ Seems like, $$x_0 + \dfrac{a_0}{2}t^2$$ is constant. How to find constant is equal to what?
  38. Safinaz

    Integration of an exponential function

    My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!
  39. bob012345

    I Exact Integration of Newton's Gravitational Law?

    I realized I never actually derived the kinematic equations of motion for the exact Newtonian gravitational force. For an object falling near the surface of the earth, how do we handle integrating the equation of motion to derive the kinematics equations without using the approximation of...
  40. M

    Monte-Carlo integration does not work properly after implementing pdfs

    Does anyone have experience with such strange behavior in Monte-Carlo methods? I think it is a conceptional problem and I am just missing a key point in how to set up the integration instead of a error in my code itself. I use data files from LHAPDF and also checked that my variables give the...
  41. DaalChawal

    MHB Integration Doubt: Answers & Solutions

  42. Eclair_de_XII

    Converting integration of rectangular integral to spherical.

    I'm going to type out my LaTeX solution later on. But in the meantime, can anyone check my work? I know it's sloppy, disorganized, and skips far more steps than I care to count, but I'd very much appreciate it. I'm not getting the answer as given in the book. I think I failed this time because I...
  43. R

    Calculating Potential Energy from Force for Non-Linear Systems

    If I have a force that behaves according to the formula ##F(x)=\alpha x-\beta x^3##, how can I get the potential energy from it? I know that: $$-\frac{\mathrm{d}V(x)}{\mathrm{d}x}=F(x),$$ but what about the limits of the integration?
  44. J

    MHB Integration Help: Struggling with Distance Qn

    I’ve always struggled with integration and I don’t know how to do this question, I’m not sure what I’m being asked to calculate. I tried to calculate this as a definite integral but there is no boundary conditions for the distance the object has traveled which is confusing any help would be...
  45. A

    Integration by filaments or integration by strate?

    Greetings While solving the following exercice, ( the method used is the integration by filaments and I have no problem doing it this way) here is the solution My question is the following: I want to do the integration by strate and here is my proposition is that even correct? I would like...
  46. A

    Changing the order of integration

    Greetings! As mentionned my aim is to change the order of integral, and I totally agree with the solution I just have one question: as you can see they have put 0<=y<=1 and 0<=x<=y^2 but would it be wrong if I put 0<=y<=1 and y^2<=x<=1? Thank you!
  47. C

    Using params from gsl function in integration routine

    I'm trying to pass through some parameters of a function to the gsl integration routine but my code is currently not returning correct values. I attach a version of my code using dummy example functions and names. struct myStruct_t { double a; }; double func(double z, void* params)...
  48. JD_PM

    Rewriting a given action via integration by parts

    I simply plugged \phi = \phi_0 (\eta) + \delta \phi (\eta, \vec x) into the given action to get \begin{align} S &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi^2 -(\nabla \phi)^2\right)-a^4V(\phi) \right] \nonumber \\ &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi_0^2 + (\delta...
  49. greg_rack

    Volume of a solid of revolution around the y-axis (def. integration)

    First, I calculated the inverse of ##y=e^x## since we're talking about y-axis rotations, which is of course ##x=lny##. Then, helping myself out with a drawing, I concluded that the total volume of the solid must've been: $$V=\pi\int_{0}^{1}1^2 \ dy \ +(\pi\int_{1}^{e}1^2 \ dy \ - \pi...
  50. E

    Calculus Practical reference for integration on manifolds

    I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity. Tong's notes and some others are good on the abstract/theoretical side but it'd really be better at this stage to get some practice with concrete examples in order to...
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