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. N

    B Does U-Substitution Function as the Inverse of the Chain Rule?

    Can someone please give as simple an example as possible to show what U substitution is about? I know basic integration rules but don't understand the point of u-substitution. I've read that it's used to "undo the chain rule", but I don't see how, and don't see how we can spot when we'd need to...
  2. M

    I Question about this Integration by Substitution

    This is part of the working from f(3x^2-1)^2xdx; I don't understand from when 6x becomes 1/6
  3. S

    Finding limit using integration

    I want to ask why the answer is not zero. If n approaches infinity, it means each term will approach zero so why the answer is not zero? Thanks
  4. Andy Resnick

    B Optimizing Exposure Times: Balancing Efficiency and Image Quality

    I'm hoping there's a reasonable answer to this. To summarize, data I acquired when imaging a particular target shows that I can retain 75% of my images for stacking at 10s exposure times, but only 50% of the images taken with 15s exposures. The difference is entirely due to tracking error and...
  5. LightPhoton

    How to turn partition sum into an integral?

    In, *An Introduction to Thermal Physics, page 235*, Schroder wants to evaluate the partition function $$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$ in the limit that $kT\gg\epsilon$, thus he writes $$Z_{tot}\approx\int_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}\,dj$$ But how is this...
  6. R

    I Deriving e=mc^2, how is it possible?

    I was recently very surprised when I had a looked up relativistic kinetic energy. All sources gave the kinetic energy as the difference between total energy and rest energy, in some or other variant of the formula ##E_k=(\gamma−1)mc^2##. I didn't really understand at first. It seemed overly...
  7. T

    I Energy of moving Sine-Gordon breather

    Hello everyone, A few days ago I stumbled across the formula for the energy of a moving breather for the Sine-Gordon equation $$\Box^2 \phi = -Sin(\phi) $$ The energy in general is given by (c=1) $$ E = \int_{-\infty}^{\infty} \frac {1} {2} ((\frac {\partial \phi} {\partial x})^2+ (\frac...
  8. MatinSAR

    Can I Use Substitution to Prove the Residue Theorem Integral?

    Integral 7.2 is ok. I must employ the integration technique used in 7.2 to prove that integral equation 7.1 equals zero. For n<0 we have : $$\sum_{n=- \infty}^{-2} a_n \oint (z-z_0)^ndz$$For n>0 we have : $$\sum_{n=0}^{\infty} a_n \oint (z-z_0)^ndz$$ According to Cauchy's Integral Theorem...
  9. L

    Double integration - switching limits

    I get that the bottom answer isn't a constant - but does this physically represent anything? When I set the two answers equal to each other, I get x = +- 1/sqrt(2) and I am wondering if this represents anything significant. I don't think (mathematically) there is anything wrong with the bottom...
  10. A

    Volume of Static Solid Using Cross-Sectional Area (Integration)

    Ok, so doing this one direction, with the range of x (0 to 2), I get the top minus the bottom equation of: ## y = 8 - x^3 ## Then, since it's squares, this would make it ##y^2##. So, integrating gives: ## \int_{0}^{2} (8-x^3)^2 = 82.3 ## That seems to be correct. However, I want to make...
  11. T

    Calculate the area of this pond with functions given for the perimeter

    So the solution is obviously given here, I'm just trying to understand it. I thought that integrating f(x) from -5 to 5 would give the area under the curve (including the areas below the "pond" at the edges of the image but above y=0. I don't really understand why we are subtracting the integral...
  12. K

    A Converting momentum sums to integrals in curved spacetime

    I am studying particle pair production using Parker and Toms book: Quantum Field Theory in Curved Spacetime. On page 48 they talk about converting the sum over momentum (k) into an integral. You assume boundary conditions so that k = 2*Pi*n/L, where n is an integer and L is the coordinate...
  13. MatinSAR

    Problem in integrating to find Rutherford's formula

    Could someone guide me on what change of variable was used to obtain equation 9.138 from equation 9.137? Book : Classical Dynamics of Particles and Systems 5th Edition by Stephen T. Thornton (Author), Jerry B. Marion (Author) They told us to check equation 8.38 and in that page they had...
  14. chwala

    Integration problem using inscribed rectangles

    Just went through this...steps pretty clear. I refreshed on Riemann integrals { sum of rectangles approximate area under curves}. My question is on the highlighted part in Red. The approximation of area under curve may be smaller or larger than the actual value. Thus the inequality may be ##<##...
  15. Manish_529

    Change in the unit vectors

    i tried integrating the stuff but it didn't work what to do
  16. Z

    Partial fractions with complex linear terms

    I am interested specifically in solving this problem by factoring the quadratic term into complex linear factors. $$s^2+4=0$$ $$\implies s=\pm 2i$$ $$\frac{5s+6}{(s-2i)(s+2i)(s-2)}=\frac{A}{s-2i}+\frac{B}{s+2i}+\frac{C}{s-2}$$ We can solve for ##C## using the cover-up method with ##s=2## to...
  17. Z

    Determine limits of integration in double integral change of variables

    $$h(t)=f(t)*g(t)=\int_0^t f(\tau)g(t-\tau)d\tau=\int_0^t g(\tau)f(t-\tau)d\tau\tag{1}$$ The Laplace transform is $$H(s)=\int_0^\infty h(t)e^{-st}dt=\int_0^\infty\left ( \int_0^t g(\tau)f(t-\tau)d\tau\right )e^{-st}dt\tag{2}$$ The Laplace transforms of $f$ and $g$ are $$F(s)=\int_0^\infty...
  18. C

    B Question about the limits of integration in a change of variables

    Hello everyone, If I have an integral ##\int_0^r \sqrt{(r^2 - x^2)}dx## and I'm integrating across the first quadrant to get the area of the first quater of a circle. And I change variables with ##x = r\cos{\theta}## and ##dx = -{r}\sin{\theta}{d\theta}## And I form a new integral that's...
  19. S

    Help with the Separation of Variables and Integration

  20. F

    Is an API Always Necessary for Server-Client Communication?

    It is clear that a server and a client are programs communicative iwth each other using one or more protocols (HTTP, TCP, etc.) I conceptually understand what an API is: it is like an intermediary between two programs that makes integration easy. For example, we build app A and want to connect...
  21. Y

    Mathematica table integration error

    I am calculating the temperature distribution and utilizing the obtained results to calculate the current distribution. In order to do this , I employ a table in which I stock all the current distribution for each value of radius . Subsequently, I aim to identify the radius corresponding to a...
  22. M

    I Help please integrating this function over a rectangular area

    Hi I struggle with integration generally. Could you be able please to talk me through the stages of this one? thanks martyn
  23. A

    Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

    I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ Which leads...
  24. J

    I Integration with different infinitesimal intervals

    Some sources state a similar format of the following $$\int_a^{a+da}f(x)dx=f(a)da$$ Which had me thinking whether the following integration can exist $$\int_a^{a+dx}f(x)dx=f(a)dx$$ I have difficulty grasping some aspects about these integrations 1. Regarding the 1st integration, shouldn't ##a##...
  25. chwala

    Integration of functions of form ##\dfrac{1}{ax+b}##

    This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?
  26. Delta2

    How Does Integration by Factors Relate to the Product Rule and FTC?

    I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself. Any helpfull tips?
  27. tehsportsmaen

    I Two-Mass Oscillator: Plotting Amplitudes Over Frequency in Hertz

    Idea: Given a system of two coupled oscillators in which 2 masses are connected to a spring in the middle. Each of the two masses is coupled to another spring on the left and right, which have fixed ends but are not connected to each other. So we have 3 springs, two masses and the springs also...
  28. G

    Help me prove integral answer over infinitesimal interval

    In the book, I see the following: ##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##. My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...
  29. MatinSAR

    Force field in spherical polar coordinates

    Picture of question: Part (a) : ##\nabla \times \vec F = 0## so a Potensial exists. I don't have problem with this part. Part (b) : what I've done : First experssion is 0 because ##\theta = \dfrac {\pi} {2}##. I don't know how to integrate over ##\theta ## when it is a constant.
  30. G

    I Integrating 1/x with units (logarithm)

    Hi. What exactly is happening mathematically when you integrate ##\frac{1}{x}## $$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$ if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...
  31. S

    Limit and Integration of ##f_n (x)##

    My attempt: (a) I don't think I completely understand the question. By "evaluate ##\lim_{n\to \infty f_n (x)}##", does the question ask in numerical value or in terms of ##x##? As ##x## approaches 1 or -1, the value of ##f_n (x)## approaches zero. As ##x## approaches zero, the value of ##f_n...
  32. Hamiltonian

    Finding a conditional probability from joint p.d.f

    using the equation mentioned under Relevant Equations I can get, $$\mathbb{P}(2X > Y |1 < 4Z < 3) = \frac{\mathbb{P}(2X>Y, 1<4z<3)}{\mathbb{P}(1<4z<3)}$$ I can find the denominator by finding the marginal probability distribution, ##f_{Z}(z)## and then integrating that with bounds 0 to 1. But I...
  33. Z

    Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##

    The characteristic equation has a zero discriminant and the sole root of ##-1##. The general solution to the associated homogeneous equation is thus $$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$ Now we only need to find one particular solution of the non-homogeneous equation. The righthand side of the...
  34. I

    A Integration of trigonometric functions

    Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
  35. Franyer

    I Integrate time dilation with derivative

    I need to integrate time dilation with derivative, how could I do that?
  36. P

    This integration appeared in the reconstruction of cross section

    I am reading the Horatiu Nastase's Introduction to quantum field theory (https://professores.ift.unesp.br/ricardo.matheus/files/courses/2014tqc1/QFT1notes.pdf ) ( Attached file ) or Peskin, Schroeder's quantum field theory book, p.105, (4.77). Through p.176 ~ p. 177 in the Nastase's Note, he...
  37. Valour549

    Two ways of integration giving different results

    I am trying to do the double integral. And I remembered there's this formula that says if the integrand can be split into products of F(x) and G(y) then we can do each one separately, then take the product of each result. Taken from Stewart's Calculus 9E. So I tried to do the integral two...
  38. E

    Work done for round trip when force is function of velocity

    I'd have no problem with this sort of problem if the force were a function of position. But here, I'm not sure where to go. Perhaps I'd start with an expression for the work done over an arbitrary distance if the force is given by ##g(v)##:$$W = \int_a^b g(v) \, dx$$ Not sure what to do next...
  39. azizlwl

    I'm getting the wrong answer for the Indefinite Integral of: (x^2+2x)/(x+1)^2

    ((x+1)^2 -1)/(x+1)^2 dx 1-1/(x+1)^2 dx Let u=x+1 1-1/u^2 du u+1/u +c (u^2+1)/u +c Not as answer given in the book.
  40. H

    Question re: Limits of Integration in Cylindrical Shell Equation

    I have managed to get the answer given by the textbook I'm referencing: 3π (∛4) (1 + 3∛3) However, this took multiple attempts, as I was initially trying to integrate within domain x = 0 - 2. This is the area for the bit that's above the x-axis (y=0 as specified). But the above answer is...
  41. chwala

    I Integration of ##e^{-x^2}## with respect to ##x##

    My first point of reference is: https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2 I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial...
  42. chwala

    Differentiate the given integral

    My take: $$\int_{x^2}^{2x} \sin t \, dt$$ using the fundamental theorem of calculus we shall have, $$\int_{x^2}^{2x} \sin t \, dt=-2x \sin x^2 +2 \sin 2x$$ I also wanted to check my answer, i did this by, $$\int [-2x \sin x^2 +2 \sin 2x] dx$$ for the integration of the first part i.e...
  43. TGV320

    I Why Are There Different Forms of the Integration Formula for Cosecant?

    Hi I have a question about the integration formula of cosecant which leaves me puzzled. I usually find it written as " = ln |csc x - cot x| + C" in most manuals, but sometimes it is written as "= - ln |csc x + cot x| + C" or "= - ln (csc x + cot x) + C". Why is that? Can they all be...
  44. chwala

    Solve the given problem that involves integration

    For part (a), Using partial fractions (repeated factor), i have... ##7e^x -8 = A(e^x-2)+B## ##A=7## ##-2A+B=-8, ⇒B=6## $$\int {\frac{7e^x-8}{(e^x-2)^2}}dx=\int \left[{\frac{7}{e^x-2}}+{\frac{6}{(e^x-2)^2}}\right]dx$$ ##u=e^x-2## ##du=e^x dx## ##dx=\dfrac{du}{e^x}## ... also ##u=e^x-2##...
  45. 1

    Integration Substitution Techniques for quadratic expressions under square roots

    Hi, With respect to the techniques mentioned in point 2 and 3: Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques. Below excerpt shows problems 4 and 5 referenced in the above 2 points:
  46. S

    Solving this definite integral using integration by parts

    Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks
  47. M

    Impulse integration for a Tennis Racket hitting a Tennis Ball

    For this, Can someone please tell me why they integrate the impulse over from ##t_i## to ##t_f##? Why not from ##j_i## to ##j_f##? It seems strange integrating impulse with respect to time. Many thanks!
  48. A

    B Explain Integration to me, please

    Hi all, I understand what the integral does - it calculates the area under a curve and can easily see how it could be used to calculate an area of land. What I do not understand is really the physical meaning when it comes to the real world. Here are some examples: 1. A set of data...
  49. F

    Insights Can You Solve the Mathematical Mystery Behind the Art of Integration?

    This article cannot replace the 1220 pages of the almanac Gradshteyn-Ryzhik but it tries on 1% of the pages to summarize the main techniques. Continue reading...
  50. kirkr

    Solution to Schwarzschild Equation for Constant t,r

    In 1916, Karl Schwarzschild was the first person to present a solution to Einstein's field equations. I am using a form of his equation that is presented in Tensors, Relativity and Cosmology by Mirjana Dalarsson and Nils Dalarsson (Chapter 19, p.205). I am approaching what may be the simplest...
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