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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2
votes
1answer
82 views

Solving the rocket differential equation

I'm trying to derive the rocket equation. I'm pretty sure that the differential equation for the rocket equation is $$v(t)\delta t =\frac{m(t)\delta t }{m(t)} V_e$$ where $v(t)\delta t$ is the ...
0
votes
1answer
39 views

Follow up on understanding path integral measures

A while ago I asked the following question: Understanding Measure in Path integrals and got to the conclusion that path integral measures are infinite products of $d\phi(x_i)$ for some scalar field $\...
3
votes
2answers
177 views

Problem with loop Integral (HQET)

I have come across the Integral: $$ \int_0^{\infty}dx [x^2-ixa+c]^{n-\frac{d}{2}}e^{-bx},$$ where $n = 1,2 ; a,b,c,d \in \mathbb{R}; b,d > 0$. This integral should contain some divergences for $d ...
1
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0answers
35 views

What is the magic behind Sector Decomposition?

I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177. So far I played around with a toy example and applied the Sector Decomposition method to ...
0
votes
1answer
66 views

Problem with converting Integral to Gamma functions (from HQET heavy quark self-energy diagram)

In the calculation of HQET radiative correction, I came across the Equation: $$\int_0^{\infty}d\lambda ~ \lambda^{-\epsilon}(\lambda+\omega)^{-\epsilon} = \frac{1}{2\sqrt{\pi}}\Gamma(\epsilon-\frac{1}{...
1
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1answer
63 views

How to find gravity field of a solid square body?

I was programming gravity simulation and stumbled upon a problem, that Newton's formula for point masses is not enough for me, I need gravity field formula of a solid square body (2D). To simplify ...
0
votes
2answers
55 views

Moment of Inertia equation for small volume

Below is the equation of the moment of inertia for small volume elements, $\Delta m$ $$I = \lim_{\Delta m_i \to 0} \sum_{i} r^2_i \Delta m_i = \int r^2 dm$$ Can someone please explain it to me on ...
0
votes
1answer
26 views

Scalar field and 2 types of line integrals

Consider the line integral, $\int _ c$f(x,y)$\vec dr$ , where $f(x,y)$ is a scalar field, and it is evaluvated on a curve $c $. After integration we get a vector let it be $\vec I$ . $\int _ c$f(x,...
0
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1answer
31 views

Meaning of flux 2-integral

Can someone please explain the meaning of flux 2-integral in this sentence: Mass is evaluated as a flux 2-integral at the asymptotic infinity. For asymptotic infinity, I believe it is as ...
-1
votes
1answer
94 views

Calculating power with force as a function of time [closed]

While doing an AP question in Physics C today I answered the question differently from the professor but I'm not sure what part of my reasoning is incorrect. A $100\ \rm{kg}$ block is being pulled ...
4
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1answer
56 views

What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?

In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as: $a(t) = a_0$ $v(t) = a_0t + v_0$ $x(t) = \frac{1}{2}...
2
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0answers
932 views

Deriving moment of inertia of a solid sphere [closed]

I have been trying to calculate it on my own, but the answer I get is different to the one I can find everywhere else, so I have to be wrong. My attempt was a very straightforward one. I used ...
-2
votes
1answer
51 views

Integral over an area of spacetime [closed]

Is it possible to evaluate this integral in spacetime? $$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$ If it is ...
0
votes
1answer
61 views

Integration Using Spherical Coordinates [closed]

So I had to find the moment of inertia of a hollow sphere of mass $M$, radius $R$, and negligible thickness. $dI=R^2 \cdot dm$ where $dm = \dfrac{M}{4\pi R^2}\cdot R^2\sin(\theta)\cdot d\theta\cdot ...
3
votes
1answer
214 views

The acceleration of the particles by finite difference [closed]

I would like to approximate the acceleration of a molecular dynamics system. I'm following an online tutorial to solve a set of equations for molecular dynamics. I can use $F=ma$ to calculate the ...
-2
votes
1answer
71 views

Given the parameters of the electrostatics problem, is this integral possible to evaluate analytically? [closed]

A cone with apex at the origin has a height $h$ and a top radius $h$, a uniform charge density with no charge on the top face. I need to find the potential $V$ at a position $z$ on the cone's axis ...
4
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2answers
325 views

Sum to an integral in deriving equipartition theorem

I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum, $${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
1
vote
1answer
66 views

Perpendicular weight force on an object that is tipping over [closed]

I'm currently working on a problem I can't seem to find an answer to. I have an object that is hanging over a cliff. This object is exactly 12m in length, and it starts off in equilibrium (6m over the ...
-4
votes
1answer
59 views

No clue about a term [closed]

$\int_S\int \vec{A}\cdot\hat{n}dS= \int_S\int A cos(\theta)dS= \int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$ I have no clue about the term $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$ ...
2
votes
1answer
121 views

Integration of the splitting function

I have a problem performing the following integration provided in the paper by Catani and Seymour (arXiv: hep-ph/9605323) page 27. Given is the integral $$ \mathcal{V}=\int_0^1 (z(1-z))^{-\epsilon} \...
1
vote
2answers
104 views

How to determine the units of my integral? [closed]

Given that I have some coefficient(i.e. a number) which is to be determined from a radial integral: $$b_{n00} = \frac{(2\pi)^{1/4}}{\sigma^{3/2}} \frac{1}{\sqrt{3}} [C(000|000)]^2 \int^{\infty}_{r = ...
1
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0answers
65 views

Integration by parts in dimensional regularisation

I have a question concerning integration by parts identities in dimensional regularisation. Appearently, almost every textbook about dimensional regularisation claims that $$\int d^Dl_1...d^Dl_L \...
1
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0answers
98 views

Explicit integration of the time-dependent Schrodinger equation when eigenvalues are unknown

Let's consider the hydrogen atom Hamiltonian $$H = - \frac{1}{2}\Delta - \frac{1}{r}$$ The solution for the corresponding time-dependent Schrodinger equation is the following: $$\psi = \psi (t = 0){...
4
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1answer
345 views

Time evolution operator in QM

I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator: \begin{align*} i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t) \end{...
0
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1answer
166 views

Integrating rigid body equations for a game engine simulation

I'm a mechanical engineer who's trying to implement a physics engine for a 3D game simulation, so I apologize for being incorrect or simply ignorant of some aspects of computation. I'm implementing ...
0
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1answer
36 views

How can I calculate the total rotation a detuned pulse will apply a nuclear spin?

I'm trying to model the effect a radiofrequency pulse will have on a nuclear spin at different detunings. The pulse has a sech lineshape, a pulse area (time integral of the pulse envelope) of $\frac{π}...
0
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1answer
41 views

How to combine limits when integrating in the frequency domain

I want to combine the signal of two separate pulses in the frequency domain in order to calculate their overlap (by multiplying the two signals together and integrating). However, one of these signals ...
1
vote
1answer
217 views

Integrating Carnahan-Starling Pressure

Given the Carnahan-Starling equation of state for a solution of hard-spheres, $$ Z = \frac{P}{\rho k_BT} = \frac{1 + \eta + \eta^2 - \eta^3}{(1-\eta)^3}$$ where $\rho = N/V$ is the number density and ...
1
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1answer
110 views

Computing integrals for divergent loop amplitudes?

I am trying to compute the cross-section for the diagram below with a divergent triangle loop: $\qquad\qquad\qquad\qquad\qquad$ where $X^0$ and $X^-$ are some fermions with zero and negative charge ...
0
votes
1answer
349 views

How to find the net electric force exerted on a uniformly charged rod by another, same rod on the x-axis (they don't touch)? [duplicate]

How to find the net electric force exerted on rod 2 by rod 1, both being on the x-axis, both having the same length and constant linear charge density, being some distance apart? More specifically, ...
1
vote
1answer
108 views

The average velocity of a particle

The Maxwell distribution of velocities is: $$p (v) = (\frac{m}{2\pi K_b T})^{\frac{3}{2}} e^{\frac{-mv^2}{2 k_b T}}$$ I want to understand how to obtain the average value of the velocity. The ...
0
votes
1answer
90 views

About the quadratures method

in the Classical Mechanics (2nd. Ed.) book of Herbert Goldstein, p. 75 it says: "Equations 3-18 and 3-20 are the two remaining integrations, and formally the problem has been reduced to quadratures..."...
1
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0answers
27 views

Hylleraas Multi-dimensional integral [closed]

Evaluate the hylleraas integral $$\int \int \frac{\exp(-ar_{1}-br_{2}-cr_{12})}{r_{1}r_{2}r_{12}}d^{n}r_{1}d^{n}r_{2} $$ with $r_{1}=|\vec{r_{1}}|$, $r_{2}=|\vec{r_{2}}|$, $r_{12}=|\vec{r_{2}}-\...
0
votes
1answer
87 views

How do I interpret this summation-integral notation?

Reading a paper I came across this abomination of a notation, my question is, how do I interpret it? For context I'll post the whole page + the notation as a separate. The notation in mind is this ...
0
votes
1answer
40 views

Integration in order to derive formulas for self-inductance

When deriving formulas for Self Inductance, I stumbled upon the following integration: $$\frac{V_o}{L}\int_0^tsinωtdt=\int_0^t\frac{dI}{dt}$$ The result logically is: $$-\frac{V_o}{ωL}cosωt +...
1
vote
1answer
381 views

Wightman Function for complex scalar field - timelike separations?

For a complex scalar field $\Phi$, the field has the expansion $$ \Phi(x^0,\mathbf{x}) = \int \frac{d^{3}\mathbf{p}}{\sqrt{ 2 E_{\mathbf{p}} (2\pi)^3 } }\ \bigg[ e^{- i E_{\mathbf{p}}x^0 + i \mathbf{p}...
0
votes
1answer
44 views

Application of Green's theorem, free electron model

I am just reading this paper about a free electron model in conjugated molecules and got a bit confused about equation (1.38). There is an integral: $$ \int_0^{a_B} f_B(x_B) \left(\frac{d}{d x_B}\...
0
votes
2answers
178 views

Derivation of the formula for Electric Potential Energy

I just learnt the formula for calculating Electric Potential Energy $W=\frac{1}{C}\int_0^Qqdq = \frac{1}{C}[\frac{1}{2}q^2]_0^Q=\frac{Q^2}{2C}$ I understand the methodology, but what I do not ...
3
votes
3answers
252 views

Peskin & Schroeder: Free particle propagation

In Peskin & Schroeder Ch. 2, p. 14, in proving that the NRQM propagation amplitude for a free particle is nonzero everywhere, they move from \begin{equation} U(t)~=~ \frac{1}{(2\pi)^3} \int d^3p \...
6
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1answer
155 views

Integration of Differential Forms

I want to understand what it actually means to integrate a differential form on a manifold. Being a mathematician, the explanation I always get is that they simply follow the right transformation rule....
1
vote
1answer
131 views

Orthonormal Basis integration and Kronecker delta

Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...
0
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1answer
91 views

What is the definition of the moment of inertia tensor?

I can find volume integrals for the moment of inertia in 2D and 3D, but is there a definition that works in an arbitrary number of (spatial!) dimensions?
1
vote
3answers
2k views

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 ...
0
votes
2answers
2k views

Derivation of Rotational Motion Equations using Calculus

How are the equations for rotational motion derived using calculus and the following general equations ? $$\mathbf{v}(t) = \mathbf{v}_0+\int_{t_0}^t \mathbf{a}(t')dt'$$ $$\mathbf{r}(t) = \mathbf{r}...
2
votes
1answer
93 views

Integrating $1/x$ in radioactive decay derivation

I have a question concerning, for example, the derivation of the equation for radioactive decay. You start with the following differential equation $$-\lambda \cdot N=\frac{\mathrm dN}{\mathrm dt}$$ ...
1
vote
1answer
965 views

Why is displacement equal to the area of velocity-time graph? [duplicate]

why is the distance of a body equal to the area of its speed-time graph? the general formula of speed(v) is v=distance(s)÷time taken(t) so the formula of distance(s) should be s=v×t so if the speed-...
4
votes
2answers
461 views

Massive versus Massless $\phi^4$ Sunset Diagram - does $\frac{1}{\epsilon^2}$ term vanish for $m=0$?

In a real scalar massive $\phi^4$-interacting theory consider the amputated sunset diagram. This is the integral out of Kleinert and Schulte-Frohlinde Critical Properties of $\phi^4$-Theories: The ...
5
votes
2answers
388 views

Moment of inertia: why $\mathrm dI=r^2\mathrm dm$ instead of $\mathrm dI=m\mathrm dr^2$?

When computing the moment of inertia, I observed that people usually use the following logic: $d I=r^2 dm,\ \therefore I=\int r^2 dm$ My question here is, why not use $dI=m ~d(r^2)$? I ...
1
vote
0answers
50 views

Closed form of Iterated integrals arising in Fredholm's integral equation solution in the context of Nonequilibrium Quantum field Theory?

While solving a Non-equilibrium quantum field theory problem I came across this class of $2n_{}^{\text{th}}$ order iterated integral : $$F(T_{}^{},T_{0}^{},\epsilon)=\int_{T_{0}^{}}^{T_{}^{}}dt_{1}^{}...
19
votes
4answers
3k views

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 ...