Integrals Definition and 1000 Threads

Homework Statement Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed. Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the...

Homework Statement [/B] F =< 2x, e^y + z cos y,sin y > (a) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along a straight path. (b) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along the curved path given by C : r(t)...

Homework Statement Hi all, could someone help me run through my work for these 2 integrals and see if I'm in the right direction? I'm feeling rather unsure of my work. 1) Evaluate ##\oint _\Gamma Z^*dz## along an anticlockwise circle of radius R centered at z = 0 2) Calculate the contour...

Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...

Homework Statement Homework Equations flux = int(b (dot) ds) The Attempt at a Solution I just wanted clarification on finding ds. I understand why ds is in the positive yhat direction (just do rhr) but I don't understand where the dxdz come from. How do we find ds in general?

Homework Statement Part b and e. Homework Equations Flux = surface integral of B (dot) ds The Attempt at a Solution I just want to make sure that I have a good understanding of ds. ds is just the direction of the vector that is normal to our area that we are finding our flux throuhg...

I have no idea if this is in the right direction. I know I am going to need the summation of the residues to use the theorem. I found the residues using the limit, but do I need to change these using the euler formula? We are supposed to be working problems at home and I am getting a bit lost...

Hello, I'm wondering if my approach to a problem is correct as when I try to simulate the DC magnetic field of a current loop using elliptical integrals, I obtain results that appear incorrect when shifting the current loop's location from the origin of the z axis. I have attached the...

Homework Statement Solve from x = 0 to x = ∞, ∫xe-axcos(x)dx Homework EquationsThe Attempt at a Solution I have a solution for the integral ∫e-axcos(x)dx at the same limits, and I feel that the result might be of use, but have no idea how to manipulate the integral above such that I can use...

I was reading a paper that gives a nice collection of all scalar integrals that crop up in QCD loop calculations. Such integrals are computed in some kinematic region and then the authors say the results may be analytically continued if so desired. I just wonder how is this analytic continuation...

Draw the regions of integration and write the following integrals as a single iterated integral. $$\displaystyle\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy + \int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$$ ok haven't done this before so kinda clueless

Taken from Emmy Noether's wonderful theorem by Dwight. E Neuenschwander. Page 28 1. Homework Statement Under what circumstances are these definite integrals functionals; a) Mechanical work as a particle moves from position a to position b, while acted upon by a force F...

Hi. I'm reading a solution to a problem concerning a gas of photons. In the solution, the energy of the gas is given as E=2\sum_{\vec{k}} \frac{\displaystyle \epsilon_{\vec{k}}}{\displaystyle \exp[\beta\epsilon_{\vec{k}}]+1} where \epsilon_{\vec{k}} is one photon's energy. It is said then...

Let S S = double integrals S S x^2 dA; where R is the region bounded by the ellipse 9x^2 + 4y^2 = 36. The given transformation is x = 2u, y = 3v I decided to change the given ellipse to a circle centered at the origin. 9x^2 + 4y^2 = 36 I divided across by 36. x^2/4 + y^2/9 = 1 I replaced...

Hello! Those who used Peskin's book on qft, in chapter 2, Causality (2.4) there are 2 integrals for ##D(x-y)##. Can someone explain to me how does he solve them, as I tried for a bit and didn't manage to do them (actually to get the behavior as ##t \to \infty##). Thank you!

I understand that if you have a function in which you want to determine the full (i.e. account for positive and negative values) integral you need to break down your limits into separate intervals accordingly. Is there any way in which you can avoid this or is it mathematically impossible? If...

<Moderator note: Merger of two threads on the topic.> Hello! I am reading some basic stuff on complex integrals using branch cuts and i found the problem in the attachment. I am not sure I understand why the branch cut is along ##R^+##. I thought that branch cut is, loosely speaking, a line...

I am really struggling setting up triple integrals. I need steps, simple steps normally applied when setting up integrals given a specific region.

So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...

I want to prove ##\displaystyle U = {1\over 8\pi}\int \vec E \cdot \vec E dV## and ##\displaystyle U = \frac12 \int \phi \rho dV## are equal. I started with ##\nabla \cdot (\phi \nabla\phi) =(\nabla \phi)^2 + \phi \nabla^2 \phi##' Then ##\displaystyle {1\over 8\pi}\int \vec E \cdot \vec E dV...

Sorry if this is the wrong place to post this, but I wasn't sure where exactly to put it. When we calculate the force on a closed loop of current-carrying wire in a uniform magnetic field, We calculate the line integral of the loop to be 0. However, when we evaluate the line integral for an...

Evaluate the double integral by converting to polar coordinates. Let S S be the double integral symbol S S xy dydx Inner limits: 0 to sqrt{2x - x^2} Outer limits: 0 to 2 The answer is 2/3. I know that x = rcosϴ and y = rsinϴ. S S rcosϴ*rsinϴ r drdϴ. S S (r^3)cosϴ*sinϴ drdϴ. I am stuck...

Evaluate the iterated integral by converting to polar coordinates. Let S S = double integral symbol S S y dx dy The outer integral is from 0 to a. The inner integral is from 0 to sqrt{a^2 - y^2}. I started by letting y = r sin ϴ S S r sinϴ dxdy. I then let dxdy = r dr d ϴ S S r sin ϴ rdr...

I am looking at Appendix A Equation 52 (Loop Integrals and Dimensional Regularization) in Peskin and Schroeder's book. ∫ddk/(2π)d1/(k2 - Δ)2 = Γ(2-d/2)/(4π)2(1/Δ)2-d/2 = (1/4π)2(2/ε - logΔ - γ + log4π) Can somebody explain how this equation is derived? I would also like to know what the...

I must evaluate the following double integral over the region R. I do not understand the limits of integration given the following equations. Let S S = double integral symbol S S x dx dy Limits of integration for x: From (4y/3) to sqrt{25 - y^2}. Note: Why is the variable y in the radicand...

Didn't know where to put this question. but just wanted to ask quickly. All i know is that in Calculus 3, you use double, triple or multiple integrals for 3D models. But what other higher level math course uses or involves multiple integrals?

Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...

I have been trying to teach myself quantum optics for some time. Up to now, I have often looked at certain types of integrals -- the ones that have operators within them -- without going into too much detail, just trying to get the general purport and moving ahead, only to get mired in some...

I am looking at a proof from a book in fluid dynamics on time differentiation of fluid line integrals - Basically I am looking at the second term on the RHS in this equation $$ d/dt \int_L dr.A = \int_L dr. \partial A / \partial t + d/dt \int_L dr.A$$ The author has a field vector A for a...

I just started learning double integrals. It is interestingly difficult. I know that switching dxdy to dydx can simplify the integration. I am not too clear why switching dxdy to dydx or vice-versa can make things easier. Let S = integral symbol SS [x/(1 + xy)] dxdy Which is easier: SS [x/(1...

I am asked to evaluate the following integral: ##\displaystyle \int_0^{ \infty} \log \left(1+ \frac{a^2}{x^2} \right) dx##. So of course this is an improper integral, but I am confused about how to go writing out the integral. From previous courses, I know that you should split the integral so...

For two improper integrals, my textbook claims that ##\displaystyle \int_0^3 \frac{dx}{(x-1)^{2/3}} = 3(1+2^{\frac{1}{3}})## and that ##\displaystyle \int_0^8 \frac{dx}{x-2} = \log 3##. However, when I put these through Wolfram Alpha, the former exists but the latter does not, and it says that...

Homework Statement ##f(x) = \begin{cases} -\frac{1}{1+x^2}, & x \in (-\infty,1) \\ x, & x \in [1,5]\setminus {3} \\ 100, & x=3 \\ \log_{1/2} {(x-5)} , & x \in (5, +\infty) \end{cases}## For a given function determine the truth of the folowing statements and give a brief explanation: a) Function...

I need to make a project that integrates physics with math, involving the use of integrals to find moment inertia of areas. The theory could be read here :http://www.intmath.com/applications-integration/6-moments-inertia.php I need to make an object that applies the theory above. Can anybody...

What would tou suggest as the best resource for learning integrals? I need preferably some practical books videos or youtube channels that deal with application and problems rather than theory. Any thoughts? Thanks

Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf). $$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...

stevendaryl submitted a new PF Insights post Solve Integrals Involving Tangent and Secant with This One Weird Trick Continue reading the Original PF Insights Post.

Consider a d dimensional integral of the form, $$\int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma} \ell^{\mu}}{D}\,\,\,\text{and}\,\,\, \int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma}}{D}$$ where ##D## is a product of several propagators. One can reduce this to a sum of scalar integrals by...

I am currently reading Young & Freedmans textbook on physics as part of a university course, and I've noticed that they repeatedly represent surface integrals (which are double integrals) as single integrals. For instance, they symbolically represent the magnetic flux through a surface as: \int...

Homework Statement [/B] J(a, b, c;y)=∫aydx/√((x-a)(x-b)(x-c)), let a<b<cHomework Equations f(θ, k)=∫0θdx/√(1-k2sin2(x)), k≤1 The Attempt at a Solution This is an example from my study material, and I don't understand the first step they do. Let x=a+(b-a)t, dx=(b-a)dt Wait...what? Why? How...

hi, This is a bit embarrassing But I do not understand what the problem is with this change of variable. suppose $$\int_{0}^{\pi }sin(u)\,du = 2$$ now make the change ## sin(u)=v ## , ## du = \frac{dv}{\sqrt{1-v^2}} ## $$\int_{0}^{0}\frac{v}{\sqrt{1-v^2}} \, dv = 0$$ other example...

Homework Statement Question: To solve the integral ##\int \frac{1}{\sqrt{x^2-4}} \,dx## on an interval ##I=(2,+\infty)##, can we use the substitution ##x=\operatorname {arcsint}##? Explain Homework Equations 3. The Attempt at a Solution [/B] This is my reasoning, the function ##\operatorname...

I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...

This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...

Where , rho 1 and rho 2 are two dimensional position vectors and K is a two dimensional vector in the Fourier domain. I encountered the above Eq. (27) in an article and the author claimed that after integration the right hand side gives the following result: I tried to solve this integral but...

Hey! :o I want to check the convergence of the following integrals: $\displaystyle{\int_2^{\infty}\frac{1}{x\left (\log (x)\right )^2}dx}$ We have that: \begin{equation*}\int_2^{\infty}\frac{1}{x\left (\log (x)\right )^2}dx=\lim_{b\rightarrow \infty}\int_2^b\frac{1}{x\left (\log (x)\right...

Hey! :o I want to check if the following integrals converge or not. $\int_0^{\infty}e^{-x}\log (1+x)dx$ $\int_0^{\infty}\sqrt{x}\cos (x^2)dx$ Do we have to calculate these integrals or do we have to use for example Direct comparison test? (Wondering) For the second one I tried for...

Hi, I see a formula of gamma function and i have a question. (1) $$\Gamma (s) = \int_{0}^{\infty } e^{-x}\, x^{s-1} dx$$ (2) $$ x=a\, n^{p} \rightarrow dx=ap\, n^{p-1}dn$$ (3) $$\frac{\Gamma (s)}{pa^{s}} = \int_{0}^{\infty } e^{-an^{p}}\, n^{ps-1} dn$$ i understand the formula but...

Homework Statement This is problem 17 from Chapter 3 of Quantum Physics by S. Gasiorowicz "Consider the eigenfunctions for a box with sides at x = +/- a. Without working out the integral, prove that the expectation value of the quantity x^2 p^3 + 3 x p^3 x + p^3 x^2 vanishes for all the...

Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...