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.
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 ##<##...
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...
$$h(t)=f(t)*g(t)=\int_0^t f(\tau)g(t-\tau)d\tau=\int_0^t g(\tau)f(t-\tau)d\tau\tag{1}$$
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
$$F(s)=\int_0^\infty...
Hello everyone,
If I have an integral ##\int_0^r \sqrt{(r^2 - x^2)}dx## and I'm integrating across the first quadrant to get the area of the first quater of a circle.
And I change variables with ##x = r\cos{\theta}## and ##dx = -{r}\sin{\theta}{d\theta}##
And I form a new integral that's...
It is clear that a server and a client are programs communicative iwth each other using one or more protocols (HTTP, TCP, etc.)
I conceptually understand what an API is: it is like an intermediary between two programs that makes integration easy. For example, we build app A and want to connect...
I am calculating the temperature distribution and utilizing the obtained results to calculate the current distribution. In order to do this , I employ a table in which I stock all the current distribution for each value of radius . Subsequently, I aim to identify the radius corresponding to a...
I proceeded as follows
$$\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$$
Which leads...
Some sources state a similar format of the following
$$\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$$
I have difficulty grasping some aspects about these integrations
1. Regarding the 1st integration, shouldn't ##a##...
I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself.
Any helpfull tips?
Idea:
Given a system of two coupled oscillators in which 2 masses are connected to a spring in the middle. Each of the two masses is coupled to another spring on the left and right, which have fixed ends but are not connected to each other. So we have 3 springs, two masses and the springs also...
In the book, I see the following:
##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##.
My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...
Picture of question:
Part (a) : ##\nabla \times \vec F = 0## so a Potensial exists. I don't have problem with this part.
Part (b) : what I've done :
First experssion is 0 because ##\theta = \dfrac {\pi} {2}##. I don't know how to integrate over ##\theta ## when it is a constant.
Hi.
What exactly is happening mathematically when you integrate ##\frac{1}{x}##
$$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$
if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...
My attempt:
(a)
I don't think I completely understand the question. By "evaluate ##\lim_{n\to \infty f_n (x)}##", does the question ask in numerical value or in terms of ##x##?
As ##x## approaches 1 or -1, the value of ##f_n (x)## approaches zero. As ##x## approaches zero, the value of ##f_n...
using the equation mentioned under Relevant Equations I can get, $$\mathbb{P}(2X > Y |1 < 4Z < 3) = \frac{\mathbb{P}(2X>Y, 1<4z<3)}{\mathbb{P}(1<4z<3)}$$ I can find the denominator by finding the marginal probability distribution, ##f_{Z}(z)## and then integrating that with bounds 0 to 1. But I...
The characteristic equation has a zero discriminant and the sole root of ##-1##.
The general solution to the associated homogeneous equation is thus
$$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$
Now we only need to find one particular solution of the non-homogeneous equation.
The righthand side of the...
Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
I am reading the Horatiu Nastase's Introduction to quantum field theory (https://professores.ift.unesp.br/ricardo.matheus/files/courses/2014tqc1/QFT1notes.pdf ) ( Attached file ) or Peskin, Schroeder's quantum field theory book, p.105, (4.77).
Through p.176 ~ p. 177 in the Nastase's Note, he...
I am trying to do the double integral.
And I remembered there's this formula that says if the integrand can be split into products of F(x) and G(y) then we can do each one separately, then take the product of each result. Taken from Stewart's Calculus 9E.
So I tried to do the integral two...
I'd have no problem with this sort of problem if the force were a function of position. But here, I'm not sure where to go. Perhaps I'd start with an expression for the work done over an arbitrary distance if the force is given by ##g(v)##:$$W = \int_a^b g(v) \, dx$$
Not sure what to do next...
I have managed to get the answer given by the textbook I'm referencing: 3π (∛4) (1 + 3∛3)
However, this took multiple attempts, as I was initially trying to integrate within domain x = 0 - 2. This is the area for the bit that's above the x-axis (y=0 as specified). But the above answer is...
My first point of reference is:
https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2
I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial...
My take:
$$\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$$
for the integration of the first part i.e...
Hi
I have a question about the integration formula of cosecant which leaves me puzzled.
I usually find it written as " = ln |csc x - cot x| + C" in most manuals, but sometimes it is written as "= - ln |csc x + cot x| + C" or "= - ln (csc x + cot x) + C".
Why is that? Can they all be...
Hi,
With respect to the techniques mentioned in point 2 and 3:
Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques.
Below excerpt shows problems 4 and 5 referenced in the above 2 points:
Using integration by parts:
$$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
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!
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...
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...
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...
I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks.
Analysis is about differentiation. Hence, complex differentiation will be my starting point...
Hello,
While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral
##
\int_0^\infty J_1(x)^2\frac{dx}{x}=1/2
##
I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
Hello all
I am trying to solve the following integral with Mathematica and I'm having some issues with it.
where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by
Where delta is a coefficient.
Due to the complex arguments I'm integrating the...
Referring to this link : https://qcdloop.fnal.gov/bubg.pdf
Using Mathematica Integrate command to solve it does not give the result stated here but I am unclear as to how they got to the result in the 4th line.
It is clear that the integrand (1st line) can diverge for certain values of the...
Ok i know that,
##\int (x+2)^2 dx= \int [x^2+4x+4] dx= \dfrac{x^3}{3}+2x^2+4x+c##
when i use substitution;
i.e letting ##u=x+2## i end up with;
##\int u^2 du= \dfrac{u^3}{3}+c=\dfrac {(x+2)^3}{3}+c=\dfrac{x^3+6x^2+12x+8}{3} +c##
clearly the two solutions are not the same...
appreciate your...
Let's assume a spaceship traveling from the Earth to the Proxima Centauri with constant acceleration g = 9.81 m/s2.
The ship is accelerating the first half of the trajectory and decelerating the second half.
I calculated the velocity profile from the Earth reference:
The travel time on...
This isn't a homework question per se but I can post more details like the data points & my work after.
Suppose we are given a set of arbitrary points for which we cannot find an equation and we need to find the area under the curve without an analytical method - we can use either of the three...
So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds.
Any help would be greatly appreciated!
Thanks!!
Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##.
Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##.
The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each...
For this problem,
The solution is,
However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q.
However, how would you solve this problem with limits of integration?
Many thanks!
I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check?
Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
Q. 3(b).
This is a textbook problem; unless i am missing something ...the textbook solution is wrong!
solution;
Mythoughts;
##f(x)=2\cos 3x-3\sin 3x## ...by using the product rule on ##\dfrac{d}{dx} (e^{2x} \cos 3x)##.
I study particle physics with “Particles and Nuclei” / Povh et al. and “Modern particle physics” / Mark Thomson and I am currently at “Deep-Inelastic scattering”. After introducing several scattering equations, such as Rosenbluth, that all include terms for electric AND magnetic scattering, i.e...