Integral Definition and 1000 Threads

. Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $ Hint: Recall that $\left\vert\displaystyle\int_a^b g(x)...

I'm looking at a proof of the asymptotic expression for the Elliptic function of the first kind https://math.stackexchange.com/questions/4064023/on-the-asymptotic-behavior-of-elliptic-integral-near-k-1 and I'm having trouble understanding this step in the proof: $$ \begin{align*} \frac{1}{2}...

Now the steps to solution are clear to me...My interest is on the constant that was factored out i.e ##\frac{2}{\sqrt 3}##... the steps that were followed are; They multiplied each term by ##\dfrac{2}{\sqrt 3}## to realize, ##\dfrac{2}{\sqrt 3}\int \dfrac{dx}{\left[\dfrac{2}{\sqrt...

R is the triangle which area is enclosed by the line x=2, y=0 and y=x. Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v## ## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...

Is the answer given above correct?

In the book it is mentioned that, in path c, the line integral would be: $$\int \vec{F}\cdot \vec{dr} = A \int_{1}^{0}xy dx = A\int_1^0 x dx = -\dfrac{A}{2}$$. but I think that dx is negative in that case, the result would be positive, right?

Find question here, My approach, using cosine sum and product concept, we shall have; ##\cos (A+B)-\cos (A-B)=-2\sin A\sin B## ##⇒\cos D-\cos C=-2\sin\dfrac{C+D}{2} \sin\dfrac {C-D}{-2}## ##⇒-3[\cos(A+B)-\cos(A-B)]=6\sin A sinB## We are given ##A=4θ## and ##B=2θ##, therefore, ##⇒-3[\cos...

I think the issue is how I parameterize my vector field, but not quite sure. In case you were wondering, this is problem # 27, chapter 16.7 of the 8th edition of Stewart. Thanks for any help.

My interest is on the highlighted part only. Find the problem and solution here. This is clear to me (easy )...i am seeking an alternative way of integrating this...or can we say that integration by parts is the most straightforward way? The key on solving this using integration by parts...

I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?

I am creating an integration technique and I have only one step left! I need to integrate ##f(x)/x^2## and then I'll be done. So I want to know if integrating this is possible. Wolfram Alpha can't integrate it, but there are problems that it couldn't solve, so I'm not 100% sure that Wolfram...

From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##. Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C. The latter ends up being 0(I calculated it parametrizing...

Hi, I have some question about evaluating a cosine function from ##-\infty## to ##\infty##. I saw for a cosine function evaluate from ##-\infty## to ##\infty## I can change the limits from 0 to ##\infty##. I have a idea why, but I can't convince myself, furthermore, is it always the case no...

I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle... So, I was curious how one might...

My attempt: Disprove. Note that ## \int_{0}^{\infty} \frac{-1 - \frac{\pi}{2} }{x^2} d x \leq \int_{0}^{\infty} \frac{\sin x+\arctan x}{x^{2}} d x ## and that ## \int_{0}^{\infty} \frac{-1 - \frac{\pi}{2} }{x^2} d x ## diverges, hence by the integral Direct Comparison Test, ## J ##...

Hi, my question regard the possibility to consider a generalization on the product integral (of type II). The product integral is defined in analogy to the definite integral where instead the limit of a sum there is a limit of a product and, instead the multiplication by ##dx## there is the...

I need help with a derivation of an equation given in a journal paper. My question is related to the third paragraph of this paper: https://doi.org/10.1007/BF00619826. Although it is about fibre coupling my problem is purely mathematical. It is about solving a complex double integral. The...

I can't understand how this approximation works ##\sum_{k=0}^m\left(\frac{k}{m}\right)^n\approx\int_0^m\left(\frac{x}{m}\right)^ndx\tag{1}##Can you please help me

I need a help in the following problem. I feel that the question is stupid. Take a function ##f\in C(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)## and a number ##\alpha\in(0,3)##. Prove that $$\lim_{|x|\to\infty}\int_{\mathbb{R}^3}\frac{f(y)dy}{|x-y|^\alpha}=0.$$ I can prove this fact by the Uniform...

I start by parametarize the surface with two variables: $$r(u,v) = (u, v, \frac {d -au -bv} c)$$ The I can get the normal vector by $$dr/du \times dr/dv$$ What limits should I use to integrate this only within the elipse? I could redo the whole thing and try write r(u, v) as u being the...

The following problem we considered with the students. Perhaps it would also be interesting for PF A homogeneous rod can rotate freely in a plane about its (fixed) center of mass O . The corresponding moment of inertia is equal to J. Two identical particles of mass m can slide along the rod...

Let ##f:[0;1)\to\mathbb{R}## and ##f\in C^1([0;1))## and ##\lim_{x\to1^-}f(x)=+\infty## and ##\forall_{x\in[0;1)}-\infty<f(x)<+\infty##. Define $$A:=\int_0^1f(x)\, dx\,.$$ Assuming ##A## exists and is finite, is it possible that ##\text{sgn}(A)=-1##?

##\int \frac{\csc{x}\cot{x}}{1+\csc^2{x}}dx## Let ##u = \csc{x}## then ##-du = \csc{x}\cot{x}dx## So, ##\int \frac{\csc{x}\cot{x}}{1+\csc^2{x}}dx## ##-\int \frac{1}{1+u^2}du = -\arctan{u} + C## ##-\arctan{\csc{x}} + C## This answer was wrong. The actual answer involved fully simplifying...

Greetings All! I have a problem finding the correct solution at first glance My error was to determine the region of integration , for doing so I had to the intersection between y= sqrt(x) and y=2-x to do so x=(2-x)^2 to find at the end that x=1 or x=5 while graphically we can see that the...

Greetings! The exercice ask to calculate the circuitation of the the vector field F on the border of the set omega I do understand the solution very well my problem is the region! I m used to work with a region delimitated clearly by two intersecting function here the upper one stop a y=3 and...

I am aware that one usually starts from the Maxwell equations and then derives the masslessness of a photon. But can one do it the other way round? The action of photon would be ##S = \hbar \int \nu (1 - \dot{x}^2) \mbox{d}t##, where ##\nu## is the frequency acting as a Lagrange multiplier...

In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows: "For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...

I am not seeing how the v goes away in the third equal sign of equation (1.8). It seems to be that it must be cos(z*sinh(u+v)), not cos(z*sinh(u)). In the defined equations (1.7), the variable "v" can become imaginary, so a simple change of variables would change the integration sign by adding...

From Feynman lectures on physics: https://www.feynmanlectures.caltech.edu/I_08.html Page 8-7 (Ch 8 Motion) “ To be more precise, it is the sum of the velocity at a certain time, let us say the ith time, multiplied by deltat. ##s=\sum_{i} v({t_i})\Delta t##” Now I suppose ##{t_i}## is some time...

Why is there a absolute value sign on sinx? Does it have to do with the domain of cot x and sin x?

Let ##u=\int_1^{u}2xdx##. \begin{align}u=& \int_1^{u}2xdx=\big[x^2\big]_1^u\\ u=&u^2-1\end{align} Which leads to ##u=\frac{1\pm\sqrt{1+4}}{2}## Assuming that the upper boundary of integration is greater than ##1##, or less than ##-1##, leads to ##u=\frac{1+\sqrt{5}}{2}\approx 1.61##. the second...

Is it possible to solve this integral? I think the substitution ##x=-u## does not help at all since it only changes variable ##x## to ##u## without changing the integrand much. Using that substitution: $$\int \frac{6x^2+5}{1+2^x}dx=-\int \frac{6u^2+5}{1+2^{-u}}du$$ Then how to continue? Thanks

Hello guys. I was trying to evaluate the integral of sine of x^2 from - infinity to + infinity and ran into some inconsistencies. I know this integral converges to sqrt(pi/2). Can someone help me to figure out why I am getting a divergent answer? $$ I = \int_{-\infty}^{+\infty} sin(x^2) dx =...

why does it say transforms? is there more than one Fourier transform?? we learned in class that the inverse Fourier transform of the Fourier transform of ##f## is ##f##, so there should be just one right? I'm uncertain of how to calulate this integral though.. Mr Wolfram showed me an indefinite...

I also don't understand how to get the descending factorials for this hypergeometric series, I also know that there is another way to write it with gamma functions, but in any case how am I supposed to do this? If I write it as a general term, wolfram will give me the result which leaves me...

Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...

I would like to solve the integral underneath: $$\displaystyle \int_{0}^{x}\!-{\frac {\lambda\,{{\rm e}^{-\lambda\,t}}{\beta}^{\alpha} \left( -\lambda+\beta \right) ^{-\alpha} \left( -\Gamma \left( \alpha \right) +\Gamma \left( \alpha, \left( -\lambda+\beta \right) t \right) \right)}{\Gamma...

$$\int \frac y {x^2+y^2} dx$$ $$\frac 1 y * \int \frac 1 {\frac {x^2}{y^2} + 1} dx = \frac 1 y * atan(x/y)$$ The answer is just atan(x/y), which you get using u-substitution but I honestly don't see why I don't get it doing it the normal way.

[FONT=georgia]Problem statement : [FONT=georgia]As a part of the problem, the diagram shows the contour ##C##above on the left. The contour ##C## is divided into three parts, ##C_1, C_2, C_3## which make up the sides of the right triangle. Required to prove : ##\boxed{\oint_C x^2 y \mathrm{d} s...

Hi there, I've been stuck on this issue for two days. I'm hoping someone knowledgeable can explain. I'm working through the construction of the quantum path integral for the free electrodynamic theory. I've been following a text by Fujikawa ("Path Integrals and Quantum Anomalies") and also...

I am trying to integrate a cross product and I wonder if the following is true. It does not feel like it is true but it would be very nice if it was since otherwise I have a problem with the signs... This is my first time posting here, so I just pasted in the LaTeX code and hope that it is...

[FONT=times new roman]Problem : [FONT=times new roman]We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##. Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using...

I am asked to compute ##[\phi(x), \phi^\dagger(y)]## , with ##\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})## and with z=x-y a spacelike vector. And show that this commutator does not vanish, which means that for this non-relativsitic field i.e. with ##p^0 = \frac{\vec{p}^2}{2m}##...

$\begin{array}{lll} I&=\displaystyle\int{\frac{dx}{x^2\sqrt{x^2-16}}} \quad x=4\sec\theta \quad dx=4\tan \theta\sec \theta \end{array}$ just seeing if I started with the right x and dx or is there better Mahalo

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Hello, To first clarify what I want to know : I read the answer proposed from the solution manual and I understand it. What I want to understand is how they came up with the solution, and if there is a way to get better at this. I have to show that, given a vector field ##F## such that ## F ...