# Questions tagged [integration]

For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

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### Electric field of an infinite sheet of charge [closed]

I am trying to derive the formula for E due to an infinite sheet of charge with a charge density of $\rho C/m^2$. I assumed the sheet is on $yz$-plane. I used Coulomb's law to get an equation and ...
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### Is dimensional analysis valid for integrals

Can we apply dimensional analysis for variables inside integrals? Ex: if we have integral $$\int \frac{\text{d}x}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right),$$ the LHS has no ...
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### Vector potential of a partially-known magnetic field

let's consider a three-dimensional space permeated by a known magnetic field $\vec{B}$. Let's consider in this space a topologically spherical surface $\mathcal{S}$ centred in the origin. I put a ...
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### Find velocity using integration method or relative velocity approach [closed]

In the diagram shown below, jeep moves with a speed of 60kmph and the car's velocity as observed from the moving jeep is 20kmph. we need to find the velocity of the car. I used relative velocity ...
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### Help with proof in Griffiths QM book [closed]

So, I'm having trouble with this proof in Griffiths' QM text. I don't get how Griffiths exactly goes from the text(circled in read) on page 47 to the next step(also circled in red). He says that he ...
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### Get rid of the derivatives and relativistic mass in Feynman lectures

i have a problem with get rid of the derivatives in Feynman lectures (chapter 15, Equivalence of mass and energy). The problem: we have $\frac {d(mc^2)}{dt} = v\cdot \frac {d(mv)}{dt}$, then we ...
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### Approximating sums as integrals and divergent terms

I have the following sum (notice that the sum starts from 2, i.e. there's no divergence): $$\sum_{i=2}^{N}C_i\dfrac{\exp{\left(-k| \mathbf{R}_i-\mathbf{R}_1| \right) }}{| \mathbf{R}_i-\mathbf{R}_1|}$$...
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### Electric field at the apex of a cone

Consider a hollow cone with uniform charge distribution over its surface. When one finds the electric field at its apex it comes out to be an infinite value. However, when a solid cone with uniform ...
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### Capacitance of tilted circular plate to ground

I am trying to calculate the capacitance between a circular plate of radius $r$ and infinite ground plane, where the circular plate is tilted at an angle $\theta$ to the ground plane. The aim is to ...
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### Solving the quantum field propagator for 1+1 dimensional spacetime at $t=0$

I just started studying Quantum Field Theory in a Nutshell By Zee to quench my thirst for physics. But I got stuck on one of the early exercises. In exercise 1.3.1 (...
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### What's the result of this integral? [closed]

$$\int_{|\vec k|<k_F} \frac{d^3k}{(2\pi)^3} e^{i\vec k\cdot \vec r}$$ it's not a Fourier transformation since the integrand is not infinite.
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### Calculation of a net electric field for a charged ring - weird integration

I am just reading book "University physics with modern physics 14-th edition (Young & Fredman)". And on page 702 there is an example 21.9 which says: Charge $Q$ is uniformly distributed ...
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### How do physicists integrate?

I've always thought that the integration notation in physics is weird, but I understood it nevertheless for a single variable, until I started reading Zee's QFT in a nutshell, where 4 dimensions are ...
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### Integration over phase space for a one-dimensional harmonic oscillator

The problem asks for a proof of the following equation, and I have no idea on how to approach this: $$\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E) ,$$ ...
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### is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
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### Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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### Issue with deriving the work-energy theorem

I'm a little confused regarding the way Total work = Change in kinetic energy is derived using calculus. My issue can be seen at 3:26 of this video: https://youtu.be/2dqO4sy4Njg?t=3m20s Why can the ...
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### Natural coordinates and time integration

So I have this physics mechanics dynamics textbook with an example and there is a step I couldn't understand in one of the solution examples. Starting with, $$mR \ddot\theta = mg\sin\theta, \tag{1}$$...
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### Gaussian integral in momentum space

My question is related to p. 353 of Altland and Simon (section 6.7) which concerns about the following field integral where $\beta = 1/T$ and $V_n$ is defined in the following way: It seems to be a ...
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### Is the formula for work $W= \vec{F}\cdot \vec{s}$ or $W=\int_C \vec{F} \cdot d\vec{s}~$?

I'm pretty much not so much introduced to calculus (I am grade 11 of India and they teach integration part of basic calculus by the end of grade 12) so I would be glad if the answer will be much more ...
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### Basic cut-off regularization

I've been reading these notes on regularization by Hitoshi Murayama here, and on page 3 there's a few lines of calculations on a quick method of regularizing an integral. But I can't follow the steps ...
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### RC Circuit Step Response Derivation

This is the step response of an RC circuit: $$V(t)=V_S+(V_0-V_S)e^{-\frac{t}{RC}}$$ Where $V(t)$ is the voltage across the capacitor with respect to time, $V_S$ is the height of the step in supply ...
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### How to derive the moment of inertia of a thin hoop about its central diameter?

For lack of a better image, I am searching for the moment of inertia of this where$\ r_1 = r_2$ (negligible thickness), and where the object would be rotating around its central diameter, which is ...
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### Integration of Acceleration to Get Delta Velocity

How do you get delta velocity if you have times t1 and t2 and their velocities v1 and v2, but you only know their accelerations a1 and a2. If you integrate over accelerations a1 and a2, do you get a "...
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### 3D volume integral and changing sign

Let $k$ be 3d momentum. What happens with $\int_{-\infty}^{\infty}d^{3}k$ when I change $k$ to $-k$? I thought that \begin{align} \int_{-\infty}^{\infty}d^{3}k&=\int_{\infty}^{\infty}dk_{x}dk_{y}...
For square integrable functions $f,g$ of a real variable, the Cauchy-Schwarz inequality states that $$\left(\int f(x)g(x)\,dx \right)^2 \le \int f(x)^2\,dx \int g(x)^2\,dx.$$ My question is: are ...