# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$ ...
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### 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{...
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### 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 ...
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### 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 ...
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### 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?
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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 ...
How are the equations for rotational motion derived using calculus and the following general equations ? $$\mathbf{v}(t) = \mathbf{v}_0+\int_{t_0}^t \mathbf{a}(t')dt'$$ $$\mathbf{r}(t) = \mathbf{r}... 1answer 93 views ### Integrating 1/x in radioactive decay derivation I have a question concerning, for example, the derivation of the equation for radioactive decay. You start with the following differential equation$$-\lambda \cdot N=\frac{\mathrm dN}{\mathrm dt}$$... 1answer 965 views ### Why is displacement equal to the area of velocity-time graph? [duplicate] why is the distance of a body equal to the area of its speed-time graph? the general formula of speed(v) is v=distance(s)÷time taken(t) so the formula of distance(s) should be s=v×t so if the speed-... 2answers 461 views ### Massive versus Massless \phi^4 Sunset Diagram - does \frac{1}{\epsilon^2} term vanish for m=0? In a real scalar massive \phi^4-interacting theory consider the amputated sunset diagram. This is the integral out of Kleinert and Schulte-Frohlinde Critical Properties of \phi^4-Theories: The ... 2answers 388 views ### Moment of inertia: why \mathrm dI=r^2\mathrm dm instead of \mathrm dI=m\mathrm dr^2? When computing the moment of inertia, I observed that people usually use the following logic: d I=r^2 dm,\ \therefore I=\int r^2 dm My question here is, why not use dI=m ~d(r^2)? I ... 0answers 50 views ### Closed form of Iterated integrals arising in Fredholm's integral equation solution in the context of Nonequilibrium Quantum field Theory? While solving a Non-equilibrium quantum field theory problem I came across this class of 2n_{}^{\text{th}} order iterated integral :$$F(T_{}^{},T_{0}^{},\epsilon)=\int_{T_{0}^{}}^{T_{}^{}}dt_{1}^{}...
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 ...