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. P

    Regarding Schwartz inequality and integration bounds

    Based on Schwartz inequality, I am trying to figure out why there should/can be the "s" variable which is the lower bound of the integration in the RHS of the following inequality: ## \left \|\int_{-s}^{0} A(t+r)Z(t+r) dr \right \|^{2} \leq s\int_{-s}^{0}\left \| A(t+r)Z(t+r) \right \|^{2} dr...
  2. Hepth

    Puiseux/Taylor Expansion of an Integrand pre-Integration

    My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...
  3. S

    Solving for Velocity: How to Integrate a Complex Function with Constants?

    Homework Statement Integrate $$v = \sqrt{2g\frac{T-v \pi r^2t}{\pi R^2}}$$ where g,T,r,R are constants Homework Equations N/A The Attempt at a Solution I tried playing around with the variables, but I am not sure how to integrate this. Just give me a little bit of hint would do. Thanks!
  4. A

    What is the electric potential at the center of a charged half-circular washer?

    Homework Statement A plastic circular washer is cut in half and has a charge Q spread uniformly over it. If the electrical potential at infinity is taken to be zero, what is the electric potential at the point P, the center of the old washer? The inner radius of the washer is a, the outer...
  5. 1

    Understanding dx in Integration

    Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually. When integrating dy/dx=x you get, ∫x dx=1/2x2. But what exactly happened to the dx, why did it...
  6. M

    Surface area of a spherical cap by integration

    hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...
  7. L

    Integration over delta distribution

    Homework Statement a) ∫δ( x² cos x) dx with x from -∞ to +∞ b) Integral over C, with C = unit circle ∫C δ(α-α0) a ds a =(1,1) Homework Equations a) δ(g(x)) =∑δ(x-xi)/|g'(xi)| b)? The Attempt at a Solution a) g(x) = x² cosx g'(x) = 2x cos x - x² sin x xi = 0 and (2n+1)π/2...
  8. N

    Calculating Average Speed: Solving the v=8sin2t+3cos2t Equation

    Hello, I was wondering if someone out there could help me. The question: The speed, v metres per second, of a particle is given as v=8sin2t+3cos2t where t is in seconds. i) Find the total distance traveled in the time interval 0≤t≤1 ii) Find the average speed over this time interval.
  9. F

    Impossible Integration involving cosx/sqrt(a-bcosx)

    Ok, I have encountered this one a few hours ago: \int \frac{cos x}{\sqrt{a - b cos x}}dx I have tried all techniques I know, integration by part, half-angle formula, substitution, trigo approach, and so on, but it seems, I could not integrate this one. I am no noob in integration, but this one...
  10. L

    Help with difficult (?) integration

    Hi everyone, I am working on a problem related to pharmacodynamics, and I'm stuck with an integral that doesn't seem to have an analytical solution. Thing is, I'm not a mathematician, so maybe there is a solution and I can't see it. I wonder if anyone can please have a look and tell me. Here's...
  11. Clarky48

    Dirac notation - expectation value of kinetic energy

    It's my first post so big thanks in advance :) 1. Homework Statement So the question states "By interpreting <pxΨ|pxΨ> in terms of an integral over x, express <Ekin> in terms of an integral involving |∂Ψ/∂x|. Confirm explicitly that your answer cannot be negative in value." ##The 'px's should...
  12. Hijaz Aslam

    Two different answers for the same integral?

    Homework Statement The anti-derivative of ∫##\frac{sinx}{sin^2x+4cos^2x}## is ##\frac{1}{\sqrt{3}}tan^{-1}((\frac{1}{\sqrt{3}})g(x))+C## then ##g(x)## is equal to : a. ##secx## b. ##tanx## c. ##sinx## d. ##cosx## Homework Equations ##d(cosx)=-sinx dx## The Attempt at a Solution I tried...
  13. mattyk

    A clarification on a step in an integration question

    Homework Statement I was given this question as a part of an assignment and lost a mark because of a step.Homework Equations the integral of cos^5(x) dx after some fiddling and substitution it gets to this (1 - u^2)^2 du In the solutions there is a step that says refine = (u^2 - 1)^2...
  14. F

    Integration where am I going wrong?

    Hi, I have an example in my studies as follows: If v = 200sin(100pi*t + 0.2) then evaluate Integral of v^2 dt between limits of 0.005 and 0. I have integrated it and used the double compound angle formula sin^2 A=1-cos2A and come up with the following as per the solution to the example in the...
  15. KostasV

    Parity and integration in spherical coordinates

    Hello people! I have ended up to this integral ##\int_{φ=0}^{2π} \int_{θ=0}^π \sin θ \ \cos θ~Y_{00}^*~Y_{00}~dθ \, dφ## while I was solving a problem. I know that in spherical coordinates when ##\vec r → -\vec r## : 1) The magnitude of ##\vec r## does not change : ##r' → r## 2) The angles...
  16. Kelly333

    Why can "dx" in integration be multiplied?

    Hi, This is an example in "Barron AP calculus" I learned from some past threads that "dx" in integration either means △x which is a infinite number or indicates the variable with respect to which you're integrating. In the equation above, it seems that dx is multiplied by (1-3x)^2. Isn't dx...
  17. F

    Odd/Even functions and integration of them

    I was not sure where to post this here or in calculus, but seeing as the underlying basic principle of my question is regarding parity of functions I am posting it here, but feel free to move if needed. Basically I am getting ready for a (intro to) QM exam and I still struggle with some basic...
  18. Rumplestiltskin

    Defined integration general question

    Not a homework problem, so not sure if this is in the right area. Thought it was too sophomore to post in the maths forums. So, when you integrate y within defined limits for x, you get the area between the graph and the x-axis within those limits for x. But a) Why is this always between the...
  19. H

    Integration of trigonometric function

    Homework Statement I have included the LaTex version of the problem. \int \frac{sin^2 x}{1+cos^2 x} dx Homework Equations Simplifying fraction Partial fractions The Attempt at a Solution I have uploaded my attempt at the solution.
  20. PhysicsBoyMan

    Area of circle using integration

    Homework Statement http://postimage.org/][/PLAIN] free picture upload 2. The attempt at a solution I want to go width times delta height. To do this I must describe width in terms of height. Here they used the Pythagorean theorem which is weird to me because I don't see a nice triangle...
  21. I

    Beam deflection integration

    Can someone explain why the slope dy/dx = 0 at x = L/2? L/2 is the midpoint and there would be a deflection here so surely the slope of the deflection curve shouldn't be 0? I'm finding it hard to visualise this. EDIT: I think I understand the above. The slope of the deflection curve at x = L/2...
  22. D

    Help with numerical integration

    I want to integrate a function numerically from 0 to infinity where for small x ##f(x)\sim x^{-5/2} \exp(-a/x)## and for large x ##f(x) \sim \exp(-bx)##. How do I best treat the steep rise for small x?
  23. Rations

    Solving for time given a messy displacement equation

    I'm having some trouble with this problem (adapted from K&K 1.23). Homework Statement An elevator is programmed to start from rest and accelerate according to \begin{align*} a(t) &= \frac{a_m}{2}\left[1 - \cos\left(\frac{2\pi t}{T}\right)\right] &\mbox{for }0\leq t\leq T \\ a(t) &=...
  24. E

    Integration application

    Homework Statement The probability that a particular computer chip fails after a hours of operation is given by 0.00005∫e^(-0.00005t)dt on the interval [a, ∝] i. find the probability that the computer chip fails after 15,000 hours of operation ii. of the chips that are still operating after...
  25. stephen8686

    Simplifying and Expanding Integration Using the Chain Rule

    Homework Statement ∫ sec2 (7-3θ) dθ Homework Equations ∫ cos(kx+b)dx= ((sin(kx+b))/k)+C ∫ sin (kx+b)dx=-cos " (k and b are constants) The Attempt at a Solution The answer in the book is: ∫ -1/3(tan(7-3x))+C. I assume by x they meant θ. The book only gave the two formulas above for solving...
  26. M

    MHB Integration by parts with absolute function

    Hi all, I have the average value of a function between limits of 7.3826 and 0 which equals 0.4453. I have used the formula for average value function and attached the equation I need solving as I don't know how to use the Latex commands. P is what I am trying to work out. Unfortunately I have...
  27. S

    Change of boundaries for an integration by substitution....

    Homework Statement Let: ##I=\int _{-1} ^{1}{\frac{dx}{\sqrt{1+x}+\sqrt{1-x}+2}}## Show that ##I=\int_{0}^{\frac{ \pi}{8}}{\frac{2cos4t}{cos^{2}t}}## using ##x=sin4t##. Hence show that ##I=2\sqrt{2}-1- \pi## Homework EquationsThe Attempt at a Solution The substitution is ##x=sin4t## which...
  28. A

    Is There a Hidden Meaning in Integrating the Lorentz Factor with Trig Sub?

    Just for my own entertainment I integrated the Lorentz factor with respect to velocity, using basic trig sub, I got the equation arcsin(v/c)*(mc^2). What does this mean? Is it just useless and irrelevant in the physics world or does it have some sort of hidden meaning?
  29. I

    Integration of the Stream Function & Velocity Potential

    How did they go from ∂Ψ/dθ = m to Ψ = mθ + f1(r) ? I mean I understand how they got mθ but can someone explain the f1(r) term? Similarly, how did they go from -∂Ψ/dr = 0 to Ψ = c1 + g1(θ) ? I understand how they got c1 but why the additional term? I can't provide further working out as...
  30. Nemo1

    MHB Calculating the area from 1280 subintervals

    Hi Community, I have this question and I stuck on the last question: (b) How many subintervals do you need to estimate the area to within 0.1units^2 I thought it was 40 subintervals but I am thinking that my answer is divisions along the $x$ axis. The other questions I am confident I have...
  31. B

    MHB Integration: Area between two parabolas on a given interval

    Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. Between y = x^2 − 5x + 2 and y = −x^2 + 5x − 6 for x in [0, 4]
  32. K

    MHB Integration Help: Finding x-coord of Point A with y= bx(x -2)

    hey guys, I am stuck on a question which is The curve in the picture shown has equation y = bx(x − 2). (a) Find b given that the shaded area is 4 units 2 (b) Find the x−coordinate of the point A if the line OA divides the shaded area into equal parts. Ive done part (a) which...
  33. D

    Double Integral: Order of Integration | 37

    Homework Statement i am done with this question and get the answer 37 but it required us to change the order of integrationHomework EquationsThe Attempt at a Solution i have no idea... i tried to do something like [/B] or but none of them can obtain the answer 37
  34. C

    Possible integration by parts?

    Homework Statement Integrate $$\int_0^1 dw \frac{w^{\epsilon+1} \ln((r+1-w)/r)}{1+r(1+w)}$$ for ##\epsilon## not necessarily an integer but positive and r is negative (<-1). The argument of the log is positive. Homework Equations Integration by parts The Attempt at a Solution [/B] I can...
  35. T

    Math problem integration by partial fractions

    Homework Statement integrate (4x+3)/(x^2+4x+5)^2 Homework EquationsThe Attempt at a Solution I know to solve this problem you have to work with partial fractions, in the solution we were given they solve as followed 4x+3=A(x^2+4x+5)'+B I don't know why they take the derivative of x^2+4x+5...
  36. squelch

    Complex Numbers and Constants of Integration

    Homework Statement Suppose that the characteristic equation to a second order, linear, homogeneous differential equation with constant coefficients yielded two complex roots: \begin{array}{l} {\lambda _1} = a + bi\\ {\lambda _2} = a - bi \end{array} This would yield a general solution of: y =...
  37. S

    Integrating Wannier Functions: Simplifying the Prefactor Equation

    Homework Statement I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear, Homework Equations w refers to wannier function, which relates to the Bloch function ##\mathbf{R}## is this case...
  38. O

    Integration Proof with constant

    Homework Statement Prove that if f'(x) = g'(x) for all x in an interval (a,b) then f-g is constant on (a,b) then f-g is constant on (a,b) that is f(x) = g(x) + C Homework Equations Let C be a constant Let D be a constant The Attempt at a Solution f(x) = antiderivative(f'(x)) = f(x) + C g(x)=...
  39. nmsurobert

    Integration Question: Confused on Limits of Integration

    im working on some homework that the instructor recently gave us the solutions to and I am confused on something that he's done. intailly i have ∫N2(x+a/2)2(x-a/2)2 the integral is from a/2 to -a/2 the next step is this ∫2N2(x2-a2/4)2 integrated from a/2 to 0 i don't understand why the...
  40. T

    Integration Help: Solve d/dt(Vc1) = Ak[c1(sat) - c1]

    Homework Statement I am working on a mass transfer problem and have this equation: d/dt(Vc1) = Ak[c1(sat) - c1]Homework Equations Initial conditions: x=0, c1=0 The Attempt at a Solution I know that the result from integration should be: c1/c1(sat) = 1 – e-(kA/V)t But I don't understand...
  41. chwala

    Differentiation and integration of implicit functions

    1. Given the function ##xy+cos y+6xy^2=0## , it follows that ## dy/dx=-y/x-siny+12xy##2. My problem is how do we integrate this derivative ## dy/dx=-y/x-siny+12xy## to get back the original function3.## ∫dy/dx dx=y ##
  42. Mr Davis 97

    Integration operation, and its relation to differentials

    I need to get a few things straight about the integration operation (as an intro calc student). I understand that integration is a process that takes a function and returns its antiderivative. We can think of it as an operator, where ##\displaystyle \int...dx## is kind of like an opening and a...
  43. J

    MHB Help With Integration Calculation - Struggling?

    \int_{-0.4088}^{-\infty}\,e^{-1/2.4^2}d struggling to solve this calculation. Not sure if i have written the formula in the right way. First post on this site. thanks
  44. kostoglotov

    Fourier, square sign wave, f(x)sin(kx) integration

    I'm not sure whether to put this here or in Linear Algebra, if any Mod feels it should go in Linear Algebra I won't mind. I've just been introduced to Fourier Series decompositions in my Linear Algebra text, and I understand all the core concepts so far from the Linear Algebra side of it (a...
  45. J

    Integration Help: Pls Explain Attached Picture

    Hi Pls can anyone explain how the attached picture was worked out? Thanks
  46. naima

    Integration by parts in curved space time

    In this thread, ramparts asked how integration by parts could be used in general relativity. suppose you have ##\int_M (\nabla^a \nabla_a f) g .Vol## Can it be written like ##\int_M (\nabla^a \nabla_a g) f .Vol## plus a boundary integration term (by integrating twice by parts)? I think thay it...
  47. W

    Conditional Expectation of Multiple Independent Random Varia

    Homework Statement Given X,Y,Z are 3 N(1,1) random variables, (1) Find E[ XY | Y + Z = 1] Homework EquationsThe Attempt at a Solution I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side and...
  48. E

    MHB Solving Double Integrals: Order of Integration Explained

    Hey guys, need your help and hope someone takes the time: I need to solve the double integral by changing the order of Integration. I would really appreciate if you could illustrate the way of how to compute the solution. Best Estelle :)
  49. N

    Process for solving integration problems

    . Homework Statement Q1. Let f(x) be any continuous function that satisfies: $$-2x≤xg(x)≤2x$$ for $$0≤x≤1$$ Find the upper and lower bounds for: $$\int_{0}^{1} \sqrt{1+g(x)+x^2}dx$$ Q2. Let h(x) be any continuous function that satisfies: $$-4≤h(x)≤x^2-4$$ for $$0≤x≤1$$ Find the upper and...
  50. T

    Triple integral changing order of integration

    Homework Statement rewrite using the order dx dy dz \int_0^2 \int_{2x}^4\int_0^{sqrt(y^2-4x^2)}dz dy dx The Attempt at a Solution I am having trouble because i don't know what the full 3 dimensional region looks like but the part on the xy plane is a triangle bounded by x = 0 , y = 4 and y =...
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