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. archaic

    B Interpreting integration otherwise

    There's one odd way to think about integration when it comes to interpreting it as a sum. suppose for a second that ##x## is in meters, you could think of distance as an infinite number ##n## of "points" in space, ##n→∞##, then in this case ##f(x)Δx## would mean that you now have ##nf(x)##...
  2. S

    I Calculus- Area between two curves (minimize it)

    Hi, This is my first question here, so please apologise me if something is amiss. I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...
  3. C

    Differential Integration Problem

    Attempt at solution: Writing the chain rule for ## E(V,T) ##: ## dE = \frac{\partial E}{\partial T}dT + \frac{\partial E}{\partial V}dV ## Then, integrating the differential: ## \int{ dE } = \int{ \frac{\partial E}{\partial T}dT } + \int{ \frac{\partial E}{\partial V}dV } ## If I put the...
  4. T

    Help with a complex integration in QFT

    N/A
  5. snatchingthepi

    Help please in understanding the limits of this integration

    So I can push this integral all the way to the end and see I get a negative volume. I solve for the intercepts of the cone and sphere at r^2 = 1/2. Seeing this cone is inside the sphere and the sphere is around it, I figure I should integrate from sqrt(1/2) to 1 since we're dealing with a unit...
  6. C

    Calculating Integrals Using the Fundamental Theorem of Calculus

    Here, width of first bar, y=x^2=a^2 y=x^2=(a+Δx)^2 height of nth bar=y=(a+(N-1)Δx)^2 Total area,I={a^2+(a+Δx)^2+(a+2Δx^2)+...+[a+(N-1)Δx]^2}Δx I={Na^2 + 2aΔx +...} I can't seem to get forward to get the required result which is 1/3(b^3-a^3)
  7. N

    Python How to combine integration equation in Python?

    I'm calculating key rate (R^Rate-wise) by integrating R(eta) over all possible eta from 0 to 1, with a probability distribution (PDTC) which is a log-normal distribution. The equation of log-normal distribution: The equation of R(eta): Therefore, R^Rate-wise =...
  8. L

    B What region in NMR spectrum should I choose for integration?

    Hi all, I have nuclear magnetic resonance spectrum. The vertical axis is intensity, and the horizontal axis is index. I need to find integral under the peak. But I am not sure, what region should I choose for integration - region 1 or region 2? Please find attached the spectrum.
  9. Dr-LucienSanchez

    Calculate the total resistance by integration using the conductivity equation

    (i) Dividing the rod into thicknesses of dx we get discs of area A with lengths=dx so using (****) we have the resistance of a typical disc (between point x' and x'+dx) as: (1) ##R(x'dx)=\frac{dx}{g(x)A}## (ii) Using (1) and (*) and the integrating from a to b of the entire rod we get...
  10. amjad-sh

    How Do You Integrate sin(2kz) from Negative Infinity to z?

    ##\int_{-\infty}^{z}sin(2kz)\,dz=\dfrac{-1}{2k} \Big [cos(2kz) \Big ]_{-\infty}^z=-\dfrac{cos(2kz)}{2k}+\dfrac{cos(-\infty)}{2k}##. I ended up here, and I don't know how to proceed. One recommended me to use contour integration, but I have no idea about it.
  11. J

    MHB Integration in Polar Coordinates (Fubini/Tonelli)

    Let $S^{n-1} = \left\{ x \in R^2 : \left| x \right| = 1 \right\}$ and for any Borel set $E \in S^{n-1}$ set $E* = \left\{ r \theta : 0 < r < 1, \theta \in E \right\}$. Define the measure $\sigma$ on $S^{n-1}$ by $\sigma(E) = n \left| E* \right|$. With this definition the surface area...
  12. Y

    Correcting Integration of tan^5x: Differentiating and Verifying the Solution

    This is my working out, and I also included the correct answer in the last line. The answer used a different method, however, what did I do wrong with my method? Thanks for the help!
  13. Gursimran Singh

    Potential energy of a shell and a disc, both covered uniformly with charge

    Double integration maybe?? I calculated potential due to shell on plate's center but not on other points on it's surface.
  14. Lardos

    A Ideas for determining the volume of a rotating object

    Hello everybody, I am currently working on an experiment investigating the formation of planets. I have a vacuum chamber in which dust particles form bigger agglomerates through accretion (sticking together). From the imagery I can see those agglomerates which are build up by smaller...
  15. M

    An integration problem using trigonometric substitution

    This is the integral I try to take. ##\int\sqrt{1+9y^2}## and ##9y^2=tan^2\theta## so the integral becomes ##\int\sqrt{1+tan^2\theta}=\sqrt {sec^2\theta}##. Now I willl calculate dy. ## tan\theta=3y ## and ##y=\frac {tan\theta}3## and ##dy=\frac{1+tan^2\theta}3## Here is where I can only...
  16. JD_PM

    What are the limits of integration for this surface integral?

    I want to compute: $$\oint_{c} F \cdot dr$$ I have done the following: $$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$ I don't know what limits the surface integral will have. Actually, I am not sure what's the surface. May you shed some light...
  17. J

    Integration of traceless symmetric matrices

    Hi, I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that $$ \int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2), $$ where ##e_{ij}## is a symmetric...
  18. looseleaf

    A Understanding Integration by Parts in Quantum Field Theory

    Hello, I'm just starting Zee's QFT in a Nutshell, I'm a bit confused about what he means by "integate by parts under the d4x". Can someone explain what he means by this? I understand how to obtain the Klein-Gordon equation from the free particle Lagrangian density, but not sure why he invokes...
  19. ju456one

    How do you calculate the work required to pump water out of a hemisphere tank?

    <Moderator's note: Moved from a technical forum and thus no template.> > The tank (hemisphere) is full of water. Using the fact that the weight of water is 62.4 lb/ft3, find the work required to pump the water out of the outlet. The radius of the hemisphere is 10. ##V =\pi x^2 h## using the...
  20. S

    Numerical integration - Gauss Lobatto

    Homework Statement I need calculate the points (##x_i##) and weights (##w_i##) with Gauss Lobatto seven points on the interval [a,b]. With the points and the weights I am going to approximate any integral at this interval.Homework Equations I have found the relevant points and weights at the...
  21. J

    MHB What is the Limit as n Approaches Infinity of the Integration of Cosine squared?

    Finding $$\lim_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(z)dz$$
  22. astrocytosis

    Volume integral over a gradient (quantum mechanics)

    Homework Statement 1) Calculate the density of states for a free particle in a three dimensional box of linear size L. 2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0## 3) Calculate the integral ##\int...
  23. M

    I Why is this volume/surface integration unaffected by a singularity?

    ##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}## Now by Gauss divergence theorem: \begin{align} \iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...
  24. T

    Attempt at volume integration to compute the full field equation

    I'm trying to figure out this volume integral, a triple integral, of a 9-variable function. 3 Cartesian-dimension variables, and 6 primed and un-primed co-ordinates. After the volume integration, the un-primed co-ordinates will have been gotten rid of, leaving a field function in terms of...
  25. J

    MHB Integration ∫ [√(sin^2 x-3sin x+2))/√(sin^2 x+3sin x+2))]dx

    Evaluation of $\displaystyle \int \sqrt{\frac{\sin^2 x-3\sin x+2}{\sin^2 x+3\sin x+2}}dx$
  26. S

    Area of a bounded region using integration

    In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
  27. Zack K

    Using Trig Substitution in Trig Integration

    Homework Statement Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$ Homework EquationsThe Attempt at a Solution I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution...
  28. Boltzman Oscillation

    I What is the integral of energy?

    So the classical law of force given by Newton is F= ma = dp/dt = qE. Thus if i integrate the last two equivalents I get: ∫(dp/dt)dt = q∫Edt p + C = q∫Edt correct? then what would the integral of...
  29. Andrea Vironda

    I Mathematical Procedure for Obtaining Velocity Profile in Laminar Flow

    Hi guys, i'm trying to find the velocity profile for a laminar flow in a round pipe. Starting from a force balance, we can obtain the first equation high in the left. I started with a procedure but i think I'm making mistakes. Can you suggest me the mathematical procedure?
  30. P

    I Integration of the Outer Product of a Basis

    Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example: The...
  31. M

    B Understanding the basics of integration

    I tried learning calculus using the book by Spivak.In this text, while introducing integrals the author explained a lot about partitioning the area under the curve and defined the integral.The way I understood this is, as we increase the number of divisions in the partition the lower sum and the...
  32. opus

    I Determining an n for Numerical Integration

    When estimating an integral using trapezoidal approximation, we can find the error or uncertainty in the estimation by: ##Error~in~T_n \leq \frac{M(b-a)^3}{12n^2}## where ##M## is the maximum value of the absolute value of f''(x) over [a,b], ##n## is the number of intervals, and ##T_n## is the...
  33. jk22

    B Usage of variable in integration

    When solving differential equations the following scripture can arise, for example: $$\int \frac{df}{\sqrt{\sin(\theta)^2-f^2}}$$ If the change of variable ##f=\sin(\theta)\sin(u)## Is performed, do the letters ##f,\theta## shall be considered independent or is...
  34. A

    I Integration being unchanged after rotation

    This question is about the general 1 loop correction to the propagator in QFT (this is actually not important for this question). Let's say we have an integral over an integration variable x, and this x ranges from ##-\infty## to ##\infty##. If we look at this integration contour in the complex...
  35. SpaceIsCool

    Can This Differential Equation Be Solved by Separation of Variables?

    Homework Statement We solved the differential equation (2.29), , for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation...
  36. N

    Integration of an analytical expression

    I have an equation regarding integration equation. Given: where is found analytically to be: My question is what is the analytical equation for equation 3? I hope that anyone may help me regarding this matter. This is the paper I referred: https://arxiv.org/pdf/1503.05793.pdf Thank you.
  37. Q

    Cylindrical Coordinates: Line Integral Of Electrostatic Field

    Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...
  38. Cathr

    I Is the Equation for Integrating an Exact Differential Correct?

    Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential. In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ## Or am I getting it wrong?
  39. N

    Python How to plot integration equation using Python?

    I have a few of integration equations and need to convert it into Python. The problem is when I tried to plot a graph according to the equation, some of the plot is not same with the original one. The first equation is the error probability of authentication in normal operation: cond equation...
  40. Eclair_de_XII

    Need help checking a quick log integration problem

    Homework Statement "Find ##\int_0^x \frac{6\ dt}{2t+1}##." Homework Equations ##\int \frac{\ dx}{x} = \ ln(x)## for ##x>0## The Attempt at a Solution Method 1: Let ##u=2t+1##, ##\frac{1}{2} du=\ dt##. ##\int_0^x \frac{6\ dt}{2t+1}=3\int_1^{2t+1} \frac{du}{u}=3\ ln(u)|_1^{2t+1}=3\ ln(2t+1)##...
  41. cromata

    I Integration over a part of a spherical shell in Cartesian coordinates

    I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system). I need to integrate this function over a part of spherical shell of radius R. This part is...
  42. Another

    Finding the Laplace transform of a piecewise function

    Homework Statement ##f(t) = -e^{-t}## ; ## t ≤ 0## and ##f(t) = 0## ;## t > 0 ## find Laplace transform this function. Homework Equations Laplace transform ##F(s) = \int_{[-∞<r<+∞]} f(t) e^{-st} dt## The Attempt at a Solution ##F(s) = \int_{[-∞<r<0]} -e^{-t} e^{-st} dt +\int_{[0<r<+∞]} (0)...
  43. opus

    Why no change in limits of integration here?

    Homework Statement Please see attached image for the full scope of the problem, and to see the work drawn out by the text. My question lies with line 3 as it is clear that u-substitution was used on a definite integral, but the limits of integration were not changed. Homework EquationsThe...
  44. R

    A Integration with Euler angle of rotation matrixes

    Hello, I was struggling with solving a specific integral. I know that I can rewrite the exponential matrices and the range of the three Euler angles. However, I am not sure I should I write in terms those three Euler angles.
  45. Delta2

    I Differentiation and Integration cannot always be swapped

    Well here is my (I hope successful this time ) attempt at 10. e) We consider the function ##f(x,y)=\begin{cases}\dfrac{e^{-x|y|}}{y}, y\neq 0 \\ 0, y=0\end{cases}##. For this function for ##y\neq 0## it is for any ##x##, ##\frac{\partial f}{\partial x}=-\frac{|y|}{y}e^{-x|y|}##. Also it...
  46. M

    Mathematica Why is the Numerical Integration Resulting in Zero for a Non-Zero Function?

    Hi PF! The following function is long but only 3 command lines: first defines the function ff, second numerically integrates the function, and third plots the function. As you'll see the integral is zero yet clearly that is not the case (seen from the plot). Any idea what's happening? ff =...
  47. K

    Determine the final result after integration

    Homework Statement acceleration of moving particle is described by a=-kv^1,5 where k is a constant. if the condition when t=0 is v=v0 and x=0 prove that xt = √(vv0).t Homework Equations dv/dt=a, dx/dt=v The Attempt at a Solution dv/dt=a dv/v^1,5=-k dt v^-1,5 dv = -k dt ← integrating both...
  48. J

    Definite Integral of Product/Composite Function Given Graph

    Homework Statement Given the graph of f(x) shown below, find the value of the integral. Photo attached. Homework Equations [/B] ∫23 5x·f(x2)dx The Attempt at a Solution [/B] I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x2))...
  49. SchroedingersLion

    Python Solving Double Integral with Monte Carlo Integration

    Greetings, I am desparately trying to solve a double integral via Monte Carlo integration. I integrate over two probability densities, the Beta distributions. Python can easily draw samples from these densities and calculate its function values. The integral can be seen here: Now my idea was...
  50. G

    Help finding ths Fourier transform

    Homework Statement find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom . Homework Equations Acos(w0t)/(d2+t2) The Attempt at a Solution I tried first to solve directly . used Euler's identity and got...
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