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. M

    Understanding Integration Limits for Spherical and Cartesian Coordinates

    Homework Statement Shown in the photo attached. 2. Homework Equations ∫V r2Sinθdθdφdr in spherical coordinates ∫V dxdydz in cartesian coordinates equation of a sphere x2+y2+z2=r2 The Attempt at a Solution In this case y=(y-2): sphere displaced on the y-axis. and since it is bound by all...
  2. F

    I Deformation of contour of integration or shifting poles

    As I understand it, in order to compute a contour integral one can deform the contour of integration, such that it doesn't pass through any poles of the integrand, and the result is identical to that found using the original contour of integration considered. However, I have seen applications...
  3. Dopplershift

    Limits of Integration of a Triangle

    Homework Statement Suppose you have a Triangle with the vertices, (0,0) (1,1) and (0,1). Integrating along that path. I have some differential function dZ where Z = Z(x,y) Homework EquationsThe Attempt at a Solution [/B] If I need to integrate, then I need to find the limits of...
  4. S

    I Solve Nonlinear DE: Friedmann Eqns for H 0-10^7

    From cosmology, the friedmann equations are given by, ##H^2 = (\frac{\dot a}{a})^2 = \frac{8\pi G}{3} \rho \, , \quad \frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho+3p) \, , \quad## where ##\rho = \frac{1}{2}(\dot \phi^2 + \phi^2)## and ##p = \frac{1}{2}(\dot \phi^2 - \phi^2)## To get ##\dot H##...
  5. Kurd

    Double integration problem for IDSFT

    Homework Statement [/B] The 2D Discrete Space Fourier transform (DSFT) X(w1,w2) of the sequence x(n1,n2) is given by, $$X(w_1,w_2) = 5 + 2j sin(w_2) + cos(w_1) + 2e^{(-jw1-jw2)}$$ determine x(n1,n2)Homework Equations By definition inverse DSFT is, $$x(n_1,n_2) = \dfrac{1}{(2π)^2}...
  6. Duke Le

    Where is wrong in this proof for rotational inertia ?

    Homework Statement Prove the formula for inertia of a ring (2D circle) about its central axis. Homework Equations I = MR^2 Where: M: total mass of the ring R: radius of the ring The Attempt at a Solution - So I need to prove the formula above. - First, I divide the ring into 4...
  7. chwala

    Solve Integration Problem: tan θ sec θ → (1/2)ln(3/2)

    Homework Statement given ## tan 2θ-tan θ≡ tan θ sec 2θ## show that ##∫ tan θ sec θ dθ = (1/2 )ln (3/2)## limits are from θ= 0 to θ=π/6 Homework EquationsThe Attempt at a Solution ##∫ tan 2θ-tan θ dθ ## → -(1/2 )ln cos 2θ + ln cos θ → ##-1/2 ln 1/2 + ln √3/2## ##= ln (√3)/2-...
  8. chwala

    Integration problem using substitution

    Homework Statement using ## u= sin 4x## find the exact value of ##∫ (cos^3 4x) dx##[/B]Homework EquationsThe Attempt at a Solution ## u= sin 4x## [/B]on integration ##u^2/2=-cos4x/4 ## , →##-2u^6={cos 4x}^3 ##...am i on the right track because now i end up with...
  9. Angelo Cirino

    I Laplacian in integration by parts in Jackson

    I am reviewing Jackson's "Classical Electromagnetism" and it seems that I need to review vector calculus too. In section 1.11 the equation ##W=-\frac{\epsilon_0}{2}\int \Phi\mathbf \nabla^2\Phi d^3x## through an integration by parts leads to equation 1.54 ##W=\frac{\epsilon_0}{2}\int |\mathbf...
  10. yecko

    Simple integration for an area problem

    Homework Statement Homework Equations Integration of graph is the area. The Attempt at a Solution I don't think my way should have any problem in it, but I can't get the right answer. Are there any careless mistakes in it? Or any other problems? And how is the true answer get? And what is...
  11. jlmccart03

    Show Wolfram Alpha's answer is equivalent to my answer.

    Homework Statement Integrate x2(2+x3)4dx. Show that Wolfram Alpha's answer is equivalent to your answer. Homework Equations No equations besides knowing that the integral of xpower is 1/power+1 * xpower + 1 The Attempt at a Solution So I have the answer to the integral by hand as (2+x3)5)/15...
  12. O

    I Epsilon-Delta or Infinitesimal: Which is More Rigorous?

    Background: mechanical engineer with a flawed math education (and trying to make up for it). I have recently read this statement (and others like it): "We shall also informally use terminology such as "infinitesimal" in order to avoid having to discuss the (routine) "epsilon-delta" analytical...
  13. T

    Integrating with respect to area? Past paper question

    This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me). The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...
  14. J

    Why Does Trig Substitution Yield Different Integral Results?

    Homework Statement ∫8cos^3(2θ)sin(2θ)dθ Homework EquationsThe Attempt at a Solution rewrote the integral as: 8∫(1-sin^2(2θ))sin(2θ)cos(2θ)dθ u substitution with u=sin(2θ) du=2cos(2θ)dθ 4∫(1-u^2)u du= 4∫u-u^3 du 4(u^2/2-u^4/4)+C undo substitution and simplify 2sin^2(2θ)-sin^4(2θ)+C The book...
  15. J

    How Is Rocket Speed and Height Calculated Under Varying Thrust and Mass?

    Homework Statement A rocket with initial mass of m0. The engine that can burn gas at a rate defined by m(t)=m0-αt, and expel gas at speed (relative to the rocket) of u(t)=u0-βt. Here, m0, α, u0, and β are all constants. Assume the lift-off from ground is immediate a) The rocket speed v(t)=...
  16. M

    Determine the surface of a cardioid

    Homework Statement Consider the cardioid given by the equations: ##x = a(2\cos{t} - \cos{2t})## ##y = a(2\sin{t} - \sin{2t})## I have to find the surface that the cardioid circumscribes, however, I don't know what limits for ##t## I have to take to integrate over. How can I know that, as I...
  17. J

    Finding the center of mass of an arbitrary uniform triangle

    Homework Statement 1. Show that for an arbitrary uniform triangle ABC, with A at (x1, y1), B at (x2,y2), C at (x3, y3), the CM (xcm, ycm), is simply defined by xcm=(x1+x2+x3)/3, and ycm =(y1+y2+y3)/3 Homework Equations xcm = 1/M * ∫xdm ycm = 1/M * ∫ydm M = ∫dm = ∫δdA where δ = M/A = dm/dA...
  18. P

    What is the Limit of e^(1/x) as x Approaches 0 and the Direction Matters?

    In this solution, in the last 3rd line, I get the first part (-e^-1 - e^-1), however, after the '-' symbol, the person writes (1/b * e^1/b - e^1/b) and takes the limit as b->0. However, shouldn't this give him (inf. * e^inf - e^inf)? Thanks (His answer is correct, by the way)
  19. J

    MHB Indefinite integration involving exponential and rational function

    Calculation of $\displaystyle \int e^x \cdot \frac{x^3-x+2}{(x^2+1)^2}dx$
  20. TeethWhitener

    I Srednicki QFT: Integration measure for KG eqn?

    Hi, I had a quick question about something from Section 3 of Srednicki's QFT book. In it, he's discussing the solution to the Klein-Gordon equation for classical real scalar fields. He gives the general solution as: $$\int_{-\infty}^{+\infty} \frac{d^3 k}{f(k)}...
  21. M

    B Why is the use of absolute value in vector norms a matter of preference?

    I would like to ask you why the author does not use absolute value of y instead of y? Source: Mathematical Methods in the Physical Sciences by Mary L. Boas Thank you.
  22. binbagsss

    Delta property, integration by parts, heaviside simple property proof

    Homework Statement I am trying to show that ## \int \delta (x-a) \delta (x-c) dx = \delta (-a-c) ## via integeration by parts, but instead I am getting ##\delta (c-a) ## (or ##\delta (a-c)## depending how I go...). Can someone please help me out where I've gone wrong: struggling to spot it...
  23. Rectifier

    Integration with variable substitution

    Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...
  24. binbagsss

    Integration of delta function over two variables

    Homework Statement I have ##\int dx \int dy \delta (x^{2}+y^{2}-E) ## [1] I have only seen expressions integrating over ##\delta## where the ##x## or the ##y## appear seperately as well as in the delta function and so you can just replace e.g ##y^2 = - x^{2} +E## then integrate over ##\int...
  25. maxhersch

    Double Integration in Polar Coordinates

    Homework Statement Integrate by changing to polar coordinates: ## \int_{0}^6 \int_{0}^\sqrt{36-x^2} tan^{-1} \left( \frac y x \right) \, dy \, dx ## Homework Equations ## x = r \cos \left( \theta \right) ## ## y = r \sin \left( \theta \right) ## The Attempt at a Solution So this is a...
  26. NoahCygnus

    Can High School Calculus Solve This Integration Problem?

    ∫ ln(e^{Φ^2}+1)dΦ I am a high school math student, so my calculus knowledge is that of high school. I tried to solve this problem, but nothing I have learned seemed to work so far, substitution didn't work, integration by parts didn't work. I presume this problem is beyond high school level...
  27. ShayanJ

    A Why can't I use contour integration for this integral?

    Consider the integral ##\displaystyle \int_{-\infty}^\infty \frac{e^{-|x|}}{1+x^2}dx ##. I should be able to use contour integration to solve it because it vanishes faster than ## \frac 1 x ## in the limit ## x \to \infty ## in the upper half plane. It has two poles at i and -i. If I use a...
  28. O

    I How Can You Integrate x-Squared Without the Fundamental Theorem of Calculus?

    OK, I admit: this will be the most idiotic question I have ever asked (maybe: there could be more) So, I am aware of the differential calculus (derivatives) and the integral calculus (integrals). And separate from that, there is the first fundamental theorem (FFT) of the calculus which relates...
  29. J

    Determining pressure at various altitudes using integration

    1. The problem statement Solve the following problems assuming air density is proportional to respective pressure at each height: What is the normal pressure at the atmosphere at the summit of a. Mt. McKinley, 6168m above sea level and b. Mt. Everest, 8850m above sea level c. At what elevation...
  30. betty0202

    Solving Boat Motor Engine Equations w/ Integration

    Homework Statement turning on the engine of a motorboat (v0=0), K = constant force due to the engine drag force of the water D = -cv find v(t)=? Homework Equations integration f=ma, a=dv/dt The Attempt at a Solution [/B] D+K = MA K-cv = MA (A=dv/dt) K-cv=Mdv/dt Mdv=dt(K-cv) ? i want to do...
  31. arpon

    Complex Integration using residue theorem

    Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
  32. egio

    I How to find outer limit of integration for this triple integ

    Here's a graph and its triple integral. How are the limits of integration for the outer integral [-2,2]? I have no idea how this was found. Any help would be appreciated!
  33. S

    I Integration by Parts without using u, v

    Hello, I'm currently taking calc 1 as an undergraduate student, and my professor just showed us a new? way of solving Integration By Parts. This is the example he gave" Is there a name for this technique that substitutes d(___) instead of dx? Thank you,
  34. chwala

    Integration of a function of x

    Homework Statement ##∫45.1/3x^2 (4-2x)^3dx##[/B]Homework EquationsThe Attempt at a Solution ##45/3∫x^2(4-2x)^3dx = let u = x^2 du= 2x, dv= (4-2x)^3 v=(2-x)/-4 ## using intergration by parts is this right[/B]
  35. T

    Work problem - Rope, pulley and brick (applied integration)

    If a brick is pulled across the floor by a rope thruogh a pulley, 1 meter above the ground - and work = W, where W = 10N , (in Newton).Show that the horizontal component of W, which is pulling the brick has the size \frac{10x}{\sqrt{1+x^2}} (*) Use this to calculate the amount of work needed...
  36. P

    B Why can't a chain rule exist for integration?

    I was thinking if the known methods of integration are enough to integrate any given function. In differentiation, we've evaluated the derivatives of all the basic functions by first principles and then we have the chain rule and product rule to differentiate any possible combination (product or...
  37. Kaura

    Taylor Series Error Integration

    Homework Statement Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4 F(x) = ∫0x sin(t^2)dt Homework Equations Rn = f(n+1)(z)|x-a|(n+1)/(n+1)![/B] The Attempt at a Solution I am...
  38. binbagsss

    Fourier transform integration using well-known result

    Problem F denotes a forward Fourier transform, the variables I'm transforming between are x and k - See attachment Relevant equations So first of all I note I am given a result for a forward Fourier transform and need to use it for the inverse one. The result I am given to use, written out...
  39. M

    Fortran Double integration over infinite intervals in Fortran

    Hi.. I am stuck up with a double integration where one of the integration limit is infinity. I know quadpack (qagi) can handle integration over infinite intervals. But how to make it work for the double integration. Or if there is any other routine that can handle both double integration and...
  40. NihalRi

    Integrating Functions with Only One Variable for Beginners

    1. Homework Statement I'm trying to integrate this, the only variable is y the others(x,w) are all constants. Homework Equations The ways of integrating that I am familiar with are substitution, trigonometric substitution, by parts & partial fraction decomposition. The Attempt at a Solution...
  41. Q

    Using Contour Integration with no singularity

    Homework Statement Using contour methods, evaluate the following integrals. In any case in which you wish to argue that some portion of a closed contour gives a negligible contribution, you should explain why that is so. Integral[E^I(k+delta*I)x^2 dx from negative Infinity to Infinity] as...
  42. S

    A How Does Dimensionality Influence a Polynomial Integral?

    Consider the following integration: $$\int \frac{d^{4}k}{(2\pi)^{4}}\ \frac{1}{(k^{2}+m^{2})^{\alpha}}=\frac{1}{(4\pi)^{d/2}} \frac{\Gamma\left(\alpha-\frac{d}{2}\right)}{\Gamma(\alpha)}\frac{1}{(m^{2})^{\alpha-d/2}}.$$ --- How does the dependence on ##d## arise in this integral? Can someone...
  43. A

    I Question about Contour Integration

    This question deals specifically with complex analysis. Let C be the unit circle in the complex plane (|z| = 1). If you calculate the contour integral of (1/z)dz over C using Cauchy's Integral Formula, you get 2*pi*i. If you calculate it using the path z(t)=e^(it), t in [0,2pi], you also...
  44. C

    Maximum Torque / Evenly Spread across a lever.

    Homework Statement The maximum torque on a lever is 1.5 x 10^6 Newtons. How many people of weight 750N can stand evenly spaced on this lever, which has a length of 20 meters? Homework Equations T=FR Weight=mg W=Fd X = Number of people The Attempt at a Solution I have set 1.5x10^6 N =...
  45. H

    Arbitrary constant in denominator

    Homework Statement Find the general solution to the differential equation: Homework Equations Separation of variables for solving 1st order separable differential equation. The Attempt at a Solution Using separation of variables, I can write: My questions are: 1) Am I correct to...
  46. nysnacc

    Integration by part not working

    Homework Statement I have three integrals, from 0 to 1 ∫ -4x5 ex3-x4dt ∫ 3x4ex3-x4dt ∫ 2tex3-x4dtHomework Equations Looks like they are not integrable, as ex3-x4 is not, I tried by part, let say u = The Attempt at a Solution
  47. tomwilliam2

    A Integration of velocity and thrust angle equation

    This is from a physics textbook, a chapter on rocket launch velocities, but really the question is how to integrate the first equation to get to the next. The way I was approaching it was like this: From ## V \frac{d\gamma}{dt}=-g \cos \gamma## Integrating from ##t=0## to some ##t##...
  48. binbagsss

    I Integration - chain rule / functional

    I have ## \int_{t = 0}^{t = 1} \frac{1}{x} \frac{dx}{dt} dt = \int_{t = 0}^{t = 1} (1-y) dt ## [1] The LHS evaluates to ## ln \frac{(x(t_0+T))}{x(t_0)} ##, where ##t_{1}=t_{0}+T## My issue is that, asked to write out the intermediatary step, I could not. I am unsure how you do this when the...
  49. S

    I Gaussian integration for complex phase

    I would like to prove that ##\displaystyle{\int dx'\ \frac{1}{\sqrt{AB}}\exp\bigg[i\frac{(x''-x')^{2}}{A}\bigg]\exp\bigg[i\frac{(x'-x)^{2}}{B}\bigg]=\frac{1}{\sqrt{A+B}}\exp\bigg[i\frac{(x''-x)^{2}}{A+B}\bigg]}## Is there an easy way to do this integration that does not involve squaring the...
  50. vmr101

    Numerical Integration of Chandrasekhar's Equation

    Homework Statement We need to write an integrator for the Chandrasekhars Equation (CE) for White Dwarfs (WD) using python3/NumPy/Matplotlib. We then need to compute the structure of a WD made of our varying elements. We also need to compute and plot the mass-radius relation for WD. Homework...
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