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. 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...
  2. 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...
  3. J

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

    Finding $$\lim_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(z)dz$$
  4. 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...
  5. 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|}...
  6. 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...
  7. J

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

    Evaluation of $\displaystyle \int \sqrt{\frac{\sin^2 x-3\sin x+2}{\sin^2 x+3\sin x+2}}dx$
  8. 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...
  9. 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...
  10. 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...
  11. 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?
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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.
  19. 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}##...
  20. 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?
  21. N

    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...
  22. 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)##...
  23. 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...
  24. 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)...
  25. 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...
  26. 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.
  27. 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...
  28. 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 =...
  29. 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...
  30. 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))...
  31. 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...
  32. 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...
  33. solarmidnightrose

    B Understanding Kinematics in Calculus for High School Students

    hi, I'm a high-school student that is just beginning to learn calculus. in calculus we are learning how to apply integration and diffrentiaiton methods regarding kinematics. there is this certain phrase i do not really understand in our textbook: e.g."in the second second" how am i meant to...
  34. N

    Electrical Potential and an Integration path

    I have been looking everywhere for a clear explanation for the following: How do I know which way to integrate the electrical field, when I'm looking for the potential between to points? Example: I have to find the potential between the top conducting plate and the bottom conducting plate...
  35. Safder Aree

    Contour Integration over Square, Complex Anaylsis

    Homework Statement Show that $$\int_C e^zdz = 0$$ Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. Homework Equations $$z = x + iy$$ The Attempt at a Solution I know that if a function is analytic/holomorphic on a domain and the contour lies...
  36. D

    I Integration of the gradient of a vector

    Hi. Is it true to say that the integral over all volume of ∇ψ where ψ is a scalar function of position and time is just ψ ? Thanks
  37. J

    Problem with Complex contour integration

    Homework Statement I want to compute ##I=\int_C \dfrac{e^{i \pi z^2}}{sin(\pi z)}##, where C is the path in the attached figure (See below). I want to compute this by converting the integral to one whose integration variable is real.Homework Equations There are not more relevant equations. The...
  38. S

    How Do You Correctly Apply Integration by Parts to ∫-e^(2x)*sin(e^x) dx?

    Homework Statement I want to integrate ∫-e^(2x)*sin(e^x) dx Homework Equations ∫uv'dx=uv - ∫u'v The Attempt at a Solution u = e^2x du = 2*e^2x dv = sin(e^x) v = -cos(e^x)/e^x e^(x)*cos(e^x) - 2∫e^(x)*cos(e^x) dx e^(x)*cos(e^x) - 2*sin(e^x) + c The solution I have doesn't have the two in...
  39. A

    Integration help on physics problem

    Homework Statement Homework Equations F=ma F/m = a The Attempt at a Solution Fc = constant force fquad= cv'^2 [/B] (Fc-cv'^2)/m = a (Fc-cv'^2)/m = dv' / dt' * using the primes to differentiate between v and v' during integration dt '(Fc-cv'^2) dv'*(m) dt' = (m/ (Fc-cv'^2)) dv' dt'...
  40. D

    I Order of integration and taking limits

    Hi. I came across the following integral in contour integration lim(ε→0) "integral of" exp(iaεeiθ) dθ = θ If I take the limit first then it just becomes the integral of 1 which is θ. I have 2 questions - If I take the limit first and then perform the integral do I always get the same answer as...
  41. ys98

    I Integrating unit vector ρ

    When doing integration such as \int_{0}^{2\pi} \hat{\rho} d\phi which would give us 2\pi \hat{\rho} , must we decompose \hat{ρ} into sin(\phi) \hat{i} + cos(\phi) \hat{j} , then \int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi , which would give us 0 instead? Thanks
  42. 9

    I Integrating Discrete Data for Navigation: A Comprehensive Guide

    I have values for the variables (C, v, g, w at all sample points) but I do not know how to evaluate the integral. This equation is supposed to be implemented on a computer as part of a larger algorithm for navigation purposes. I have a feeling that the gyroscope sensor reading and or the...
  43. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...
  44. Philip Robotic

    Integration that leads to logarithm functions problem

    Hi everyone, So I am a high school student and I am learning calculus by myself right now (pretty new to that stuff still). Currently I am working through some problems where integration leads to logarithm functions. While doing one of the exercises I noticed one thing I don't understand. I...
  45. E

    Numerical Integration in Python

    I want to find the numerical solution to the following nested integral in Python \frac{K!}{(K-M)!}\int_{x_1=0}^{y}\int_{x_{2}=x_1}^{\max(x_1,\,y-x_1)}\cdots \int_{x_M=x_{M-1}}^{\max(x_{M-1},\,y-\sum_{m=1}^{M-1}x_m)} \frac{1}{(1+x_M)^{K-M+2}}\prod_{m=1}^{M-1}...
  46. navneet9431

    B What Is the Harm in Using Finite dx in Calculus?

    What does the author want to say through these lines? What is the *harm* he is talking about? I will be thankful for any help!
  47. D

    Contour integration around a square

    Homework Statement I am trying to calculate the contour integral of the complex conjugate of z around a square with sides of length 2 centred on the origin in the complex plane Homework Equations ∫ f(z) dz = ∫ f(t) (dz/dt) dt . It looks like the integral signs won't appear here but...
  48. Krushnaraj Pandya

    Integration problem (algebraic+trigonometric function)

    Homework Statement find integral of the function- (x^2 + cos^2)(cosec^2) / (1+x^2) 2. The attempt at a solution I noticed the denominator is the derivative of Arctan(x), I tried integrating by parts with various choices for 1st and second function but all of them end up being more complicated...
  49. Krushnaraj Pandya

    Integration problem ∫1/(√3 sinx+ cosx) dx

    Homework Statement ∫1/(√3 sinx+ cosx) dx is Ans. as per textbook 1/2(log(tan((x/2)+(pi/12)) 2. The attempt at a solution Attempt 1-I changed it to half angles (x/2) and then multiplied and divided sec^2(x/2) to the numerator and denominator, then putting tan(x/2)=t I got 2∫1/(√3t+1-t^2) I wrote...
  50. Krushnaraj Pandya

    Integration problem ∫sqroot((cosx-cos^3(x))/(1-cos^3(x)))

    Homework Statement Find ∫sqroot((cosx-cos^3(x))/(1-cos^3(x))) 2. The attempt at a solution I tried taking cosx common on top which gives cosxsin^2(x), but it didn't give me any benefit, I tried splitting the numerator, factoring in sec^2(x) and a bunch of substitutions to no avail. Finally...
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