# Tagged Questions

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### Notation regarding the continuity equation for conservation of mass

I have the following equation for the net mass flow out of a control volume through a surface $S$ - $$\int \int_S p \overrightarrow{V} \cdot \overrightarrow{d}S$$ (Actually there should be an ellipse ...
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### Weird integral symbol : $\mathrel{\int\!\!\!\!\!-}$

What does this integral sign mean ($\int$ with line going through the middle)? $$\mathrel{\int\!\!\!\!\!\!-}$$ (It had something to do with the Beckenbach-Radó Theorem)
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### Find the value of $m$ given that $\displaystyle\int_0^m \dfrac{dx}{3x+1}=1$

I have to find the value of $m$ such that: $\displaystyle\int_0^m \dfrac{dx}{3x+1}=1.$ I'm not sure how to integrate when dx is in the numerator. What do I do? edit: I believe there was a typo in ...
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### Why do we need the absolute value signs when integrating the square of a function?

Why do we need the absolute value signs in the definition of square-integrable function? As seen below: $$\int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty$$
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### Integral notation

I have encountered the following integral: $\int_{x-d}^{x+d}f(y)dy$ I am trying to figure out what is the role of $d$ in this integral. Is the $d$ at the beginning of the integral the same as the ...
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### Why are variables in integration by substitution so counter intuitive?

Integration by substitution is defined as something like $\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$ But for my taste, the variables $x$ and $t$ are exactly ...
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### How do you write an integral and why

A. Year 1 Calculus Student Approach $$F(x) = \int f(x') dx\,$$ B. Random math paper you find online approach $$F(x) = \int dx f(x') \,$$ C. Spivak $$F(x) = \int f(x) \,$$ D. ??? (Edit) ...
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### Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
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### Notation for Surface Integral in $\mathbb{R}^3$

Recently, a paper of mine got accepted, but the reviewers are struggling with the (in my view) standard notation for surface integrals in $\mathbb{R}^3$: Let $\Gamma \subset \mathbb{R}^3$ be a ...
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### Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
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### Integration notation

could someone explain the following notation: $u(x,t):=\int^t_0 v(x,t:\tau)d \tau$ It's come up but I don't understand how to interpret the semicoloned tau
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### Integrating a term again and again

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to ...
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### Strange Integral Notation? [duplicate]

When reading about an certain algorithm (about parameter estimation for Kalman Filtering page 7 eq 57) I found this notation: $\int dx f(x)$ which is normally written as $\int f(x) dx$. I spent a ...
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### Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
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### Why $dt/t$ in Mellin transform

I've noticed that often when people write the Gamma function $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt$, that they write it like $$\Gamma(s) = \int_0^\infty t^s e^{-t}\,\frac{dt}{t} ,$$ where ...
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I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ... 0answers 65 views ### Integrating With Respect To$x$Suppose I have the first derivative of the function$y$,$\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function$y$by integrating with respect to$x$: ... 0answers 83 views ### Integration by parts and notation. I've just found that I must be missing something about the integral notation when it comes to the integration by parts. First,$\int_a^b \! f(x) \, \mathrm{d}x$is perfectly clear:$\mathrm{d}x$is ... 1answer 43 views ### A notational problem of a double integral What does the following double integral mean?I don't quite understand the notation. $$\int_0^1dy\int_\sqrt{y}^\sqrt{2-y^2}f(x,y)~dx~.$$ Does it equal to ... 3answers 88 views ### Prime notation in integral? Recall the definition of potential energy: $$U_x-U_{x_0} = -\int^x_{x_0}F_x(x')dx'$$ I've seen the integral definition of work, but not this - the thing I'm specifically interested in is the ... 1answer 185 views ### How can a single integral equal a triple integral? Here is part of a discussion about the gravitational potential of a sphere: Let$dxdydz$represent an infinitesimal volume containing matter of density$\rho$and mass$dm$. Then the ... 1answer 50 views ### Integrals with infinite bounds sometimes written as limits, sometimes not? When I saw Wikipedia's notation for the inverse Laplace transform, I became curious if there was a reason behind it. Is there a reason why Wikipedia writes the inverse Laplace transform as this ... 1answer 274 views ### Derivative$\Delta x$and$dx$difference This may seems like a dummy question but I need to ask it. Consider the definition of derivative: $$\frac{d}{dx}F(x) = \lim_{\Delta x->0}\frac{F(x+\Delta x) - F(x)}{\Delta x} = f(x)$$ Also: ... 1answer 100 views ### Meaning of the following integal What does$\int d^3 x $mean? I found this in a lecture on quantum field theory, and it was not explained. 0answers 66 views ### Why write$\mathrm dx, \mathrm dt$etc. at the beginning of an integral? [duplicate] I've noticed that many people here (on Math.SE) as well as elsewhere write integrals out like this: $$\int^a_b \mathrm dt \; f(t)$$ instead of the more common (at least from what I've seen): ... 4answers 315 views ### Evaluating$\int \frac{dt}{(\cos(t))^2}$? How do I solve an integral with a differential on top? E.g.: given this integral to evaluate: $$\;\int \frac{dt}{(\cos(t))^2}\;\;?$$ What does it even mean when there's a differential? 3answers 334 views ### Meaning of$\int\mathop{}\!\mathrm{d}^4x$What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this$\int f(x)\mathop{}\!\mathrm{d}x$is the integral of the function$f$over the$x$variable, but the following ... 3answers 342 views ### Understanding the differential$dx$when doing$u$-substitution I just finished taking my first year of calculus in college and I passed with an A. I don't think, however, that I ever really understood the entire$\frac{dy}{dx}$notation (so I just focused on ... 1answer 406 views ### Integral sign with circle (AND arrow on the circle) through it I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path. So when I encountered in complex analysis the above integral sign but with ... 3answers 75 views ### Different methods to write an integral I saw someone write this for showing substitution. Is it correct. $$\int \frac{2x}{\sqrt{x^2+2}}\, \mathrm{d}x$$ $$\int \frac{\mathrm{d}u}{\sqrt{u}}\, \mathrm{d}\left(2xdx\right)$$ Just wondering ... 2answers 200 views ### Evaluating using BRA KET Notation? Evaluate$\langle 0 \mid x^3 \mid 1\rangle$, assuming that all the wave functions you encounter are normalized eigenfunctions of the harmonic oscillator Hamiltonian, without Mathematica, Maple, or ... 4answers 846 views ### Why doesn't Spivak ever write$dx$in an integral? I've noticed that Spivak, and many other analysis books I read like Munkres, do not use$dx$when they integrate. Why is that? This is a serious question. 5answers 1k views ### What is this symbol$\int$called? Title says it all (I think). I'm betting it's something to do with Standard Integrals though. What is this symbol "$\int$" called? 1answer 234 views ### Integration Antiderivative vertical bar [duplicate] Possible Duplicate: What is the name of the vertical bar? When taking a definite integral, the first step is finding the anti-derivative. Once you have gone through all the steps to ... 3answers 427 views ### What does$\inf\int$mean? What does notation $$\inf\int f(x) \,\mathrm{d}x$$ stand for? I noticed it in a question on this site. Any keywords, links or stories about this or similar notations will be appreciated :) Sorry ... 0answers 236 views ### Is there a name or definition for this popular notation? I'm sorry if this is a silly question. I've done quite a bit of searching and have not found any definition or name for this symbol/usage, despite immense popularity and convenience. The sources I've ... 1answer 143 views ### “Discovering” the indefinite integral's notation I'm currently reading Keisler's Elementary Calculus -- An Infinitesimal Approach, which develops the main results usually thought in undergrad calculus using Robinson's hyperreal numbers (instead of ... 1answer 114 views ### Limit of an n-ary product Since a definite integral is defined as $$\lim_{n\to\infty} \sum_{i=0}^n f(x_i^*)\,\Delta x = \int_a^b f(x)\,dx$$ and the integral is much easier to calcluate than a sum, if we change the sum to a ... 1answer 294 views ###$\int f(x) dx $is appearing as$\int dx f(x)$. Why? [duplicate] A few of us over on MITx have noticed that$\int f(x) dx $is appearing as$\int dx f(x)$. It's not the maths of it that worries me. It's just I recently read a justification (analytical?) of the ... 1answer 101 views ### Confusion with (notation in) a surface integral problem. I encountered the following problem in a practice Advanced Calc exam, and I have an issue. Suppose$\phi:\mathbb{R}^3\to\mathbb{R}$is a strictly positive function satisfying ... 2answers 264 views ### Notation for antiderivative I came across the following: $$F(x) = \int x^3 \cos(x)dx$$ where$F$is understood to be a primitive of$x^3 \cos(x)$. I find this confusing, because of the "same"$x$appearing on both sides of ... 0answers 76 views ### Notation for application of the sequence of integrals In the paper I'm writing I often encounter expressions like $$\int dx_1 \int dx_2 \ldots \int dx_N \mathrm{<something>}.$$ In this form it is not that cumbersome, but things get worse when ... 2answers 506 views ### Necessity of Brackets for Integration Suppose I want to integrate$f(x)+g(x)$. Can this be written as$\int f(x)+g(x)\, dx$or are brackets necessary, i.e.$\int \left(f(x)+g(x)\right) \,dx$? 1answer 511 views ### This multiple integral notation, has it got a name?$\int dx \int dy \, f(y,x)$I've encountered, on Wikipedia (examples below), an integration notation which seems to be prefix-style: the integral sign is immediately followed by the$\mathrm dx$(or$\mathrm dy\$, or what have ...
In a summative assessment, I lost a mark due to this: $$f_X(x)=\frac{\lambda}{\sqrt{2\pi}}\int_{-\infty}^\infty y^2\exp\left\{-\left(\frac{1}{2}+\lambda x\right)y^2\right\} \; dy$$ Now let ...
I've seen some problems where the OP writes integrals in this form $$\int {dt} f\left( t \right)$$ or for double integrals $$\int {dx} \int {dtf\left( {t,x} \right)}$$ Do they represent another ...