Questions tagged [integration]

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

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8 views

compute $\iint_Y F.N \ dS$

The question is: Find $\iint_Y F.N \ dS $ $$ F=(x^4+yz-x^5,5x^4y,z),\quad \text{The surface }Y=x^2+y^2-z^2=1, \ \ 0\leq z\leq 1 \quad N=\text{ the normal points away from z axis}$$ Here is how i ...
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Question on marginal probability density function

Consider $\zeta \sim U[-2; 2]$, $\eta \sim U[0; 1]$, $Z = \zeta + \eta\zeta$, $\zeta$ and $\eta$ are independent. First of all, I need to find the conditional density of $Z \vert\zeta=x$. Let $\zeta = ...
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34 views

Finding $\int_{1}^{2} \sqrt{1+\frac{1}{x^2}+\frac{1}{(x+1)^2}} dx$

This integral $$\int_{1}^{2} \sqrt{1+\frac{1}{x^2}+\frac{1}{(x+1)^2}} dx$$ has been fabricated keeping an interesting point in mind, I may present a solution later. The question is: How will you do it ...
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Integrate $\int \frac{\tan \left(x\right)}{\sin ^2\left(x\right)}dx\cdot \int \log _{3x}\left(x^2\right)dx$.

Integrate the following integral: $$\int \frac{\tan \left(x\right)}{\sin ^2\left(x\right)}dx\cdot \int \log _{3x}\left(x^2\right)dx$$ I did this question a while ago and the answer is correct. ...
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22 views

Evaluate the integral using the given data

Let $f:[0,1]\to [0,1]$ be a continuous function such that $x^2 +(f(x))^2\le 1$ for all $x\in [0,1]$ and $\int_0^1 f(x).dx=\frac{\pi}{4}$, then find $\int_{-1/2}^{1/\sqrt 2} \frac{f(x)}{1-x^2}.dx$ The ...
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2answers
28 views

How to find the volume of this region?

I want to find the volume of the following subset of $\mathbb R^3$: \begin{equation} (x^2+y^2)^2\leq x,\quad 0\leq z\leq 2x-\sqrt{x^2+y^2}\text{.} \end{equation} I tried to draw a picture of the given ...
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49 views

Number of roots of the equation $f(x)= \int_0^x (t-1)(t-2)(t-3)(t-4)dt =0$

I have the following question before me: Find the number of roots of the equation $f(x)= \int_0^x (t-1)(t-2)(t-3)(t-4)dt =0$ in the interval $[0,5]$. $0$ is clearly one of the roots. But how can I ...
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27 views

Integration by subsitution

Im confused by the subsitution, I would've assumed it would be I can give more context if needed, its from the paper. On the generality of the relationship among contact stiffness, contact area, and ...
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1answer
47 views

Integral trick explanation or link

We used the following integral trick in our lecture, I really can't understand why it works. I would appreciate an explanation or even just a name so i can search for it :) So apparently: $$\int_{-\...
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24 views

Calculate the following integral in the given set

I have to calculate the Integral of the following function in the given set D. Now I went on to try to write this D as a set of the form $ E$ = {$(x,y) \in \mathbb{R^2}: a < x < b , \alpha(x)&...
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Pointwise convergence and derivatives [duplicate]

Let $f,g$ $C^0$ functions from $[0, 1] \to \mathbb R$, and a sequence of $C^1$ functions $f_n : [0, 1] \to \mathbb R$. Show that if $f_n'$ converges uniformly to $g$ and $f_n$ converges pointwise to $...
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36 views

Problem with Poisson integral

I have heat equation $$ u_t=u_xx \\ u(x,0)=xe^{\frac{-x^2}{3}} $$ I solve by the Poisson integral $$ u(x,t)=\frac{1}{2\sqrt{\pi t}} \int xe^{\frac{-x^2}{3}}e^{\frac{-(y-x)^2}{4t}} dx $$ I worked with ...
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13 views

Find the relationship between $x$ and $y$ so that $y:=0\rightarrow\frac{\pi}{2}\Leftrightarrow x:=y\rightarrow\frac{\pi}{2}.$

Find the relationship between $x$ and $y$ so that $y:=0\rightarrow \dfrac{\pi}{2}\Leftrightarrow x:=y\rightarrow \dfrac{\pi}{2}.$ I'm having trouble solving the double integral if I change the order ...
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28 views

Finding the Riemann $\zeta$ function by adelic integration

I am referring to Tao's blog post about Tate's thesis. Introduce the adeles $\mathbb A$ of $\mathbb Q$ and the adelic Mellin transform $$Z(s) = \int_{\mathbb A^\times} = g(x) |x|^s d^\times x.$$ Here, ...
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26 views

Find $f:[0, 1]\to \mathbb R$ that maximizes $I(f)-J(f)$, where $I(f)=\int_0^1 {x^2 f(x)dx}$, $J(f)=\int_0^1{x\left(f(x)\right)^2 dx}$

I've never seen this kind of problems - finding a function with almost no conditions which maximizes the integral - so I'm asking for a hint. The problem is as follows. Find a continuous function $f:[...
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33 views

Improper integrals ; Why does this equality hold?

$ H(t):= \begin{cases} \dfrac{1}{2}-t & (0<t<1) \\ 0 & (t=0) \\ \text{periodic of period 1} &(\text{otherwise}) \end{cases}$ $ \mu (x):=\sum_{n=0}^{\infty} \displaystyle\int_0^1 \...
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1answer
37 views

A p-adic Fourier transform

Consider the field of $p$-adic numbers $\mathbb Q_p$. Define the character $\chi(u p^n) = e(p^n)$ for all $n \in \mathbb Z$ and all unit $u$. In particular it is trivial on integers. This allows to ...
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84 views

Differentiate under the integral sign: $ \int_0^{2π} e^{\cos (x)}\cos(\sin x) \, \mathrm{d}x $

I have again a doubt regarding an exercise of differentiation under the integral sign. In this case, it concerns the integral: $$ \int_0^{2π} e^{\cos(x)}\cos(\sin x) \, \mathrm{d}x $$ I tried the ...
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Theorem 6.10 Rudin PMA, Partition

The third paragraph of the proof begins, 'Now form a partition $P = \{x_0,x_1,...,x_n\}$ of $[a,b]$, as follows: Each $u_j$ occurs in $P$. Each $v_j$ occurs in $P$. No point of any segment $(u_j,v_j)$ ...
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1answer
46 views

Line integral calculation for a line segment

Compute the line integral $\int_y e^z dz$ where $y$ is the line segment from $0$ to $z_0$. Since there is only one variable z here, I can directly compute $\int_y e^z dz=\int_0^{z_0} e^z dz=e^{z_0}-1$...
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32 views

Standard path for integral with complex limits?

my question is quite simple and might be a duplicate though I couldn't find one. Is there an accepted meaning for the notation $$\int_{z_1}^{z_2}$$ With $z_1,z_2\in\mathbb{C}$? Is the standard to just ...
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1answer
54 views

Particular integral for $a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=\ln x$

I would like to know how to find a particular integral for $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=\ln x$$ where $a,b,c$ are constants. So far, the only functions I've come across for $f(x)$ in $$a\...
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31 views

How to calculate this integral: $\int_{C}{} (z^2-2z) dz$ where $C$ is a line segment starting at $1$ and ending in $i$

The task is as follows: $$ \text{Calculate } \int_{C}{}f(z) dz \text{, where } \\ \text{a) } f(z)=z^2-2z \text{, } C \text{ - line segment starting at } 1 \text{ and ending at } i \text{;} \\ \text{b) ...
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1answer
25 views

Volume using triple integral with spherical and cylindrical coordinates [closed]

I had to solve this triple integral and I tried to solve by cylindrical and spherical coordinates but couldn't get anywhere. I was hoping someone could help me in this problem. Solve $\int \int \int _{...
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48 views

$\lim_{x\to a}\frac{a}{x-a}\operatorname{Si}_a(x)$

Let $a>0$ and $\operatorname{Si}_a(x)=\int_{a}^{x}\frac{\sin(t)}{t} \, dt$. Compute \begin{equation*} \lim_{x\to a}\frac{a}{x-a}\operatorname{Si}_a(x) \end{equation*} My reasoning was: suppose $F$ ...
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16 views

Fourier series transformation help - unable to reconstruct (simple) paper results

I am reading a paper and am having difficult reconstructing an equation. The paper begins with the following equations where $\omega_{n} = \omega_{o} + n\Delta\omega$ and: (1) $A_{in}(t) = \sum_{n}a_{...
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4answers
38 views

Solid of revolution axis $y=5$

the problem goes like this Rotate the indicated area around the given axis to calculate the volume of the solid of revolution $y=x^2+1$, $x=0$, $x=2$, $y=0$ , around the axis $ y = 5$ My question is ...
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34 views

How do you find the complex $(C_n)$ fourier series expansion of $e^{ax}$?

How do you find the complex $(C_n)$ fourier series expansion of $e^{ax}$? The period of the function is 2$\pi$. I've tried to do this question out several times, and I keep getting stuck and don't ...
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21 views

Variable change in multiple integrals: polar coordinates at zero.

We can make a variable change if we have some diffeomorphism $\phi$. A usual variable change is polar coordinates. But when $\rho = 0\quad |\phi'| = 0$, so $\phi$ is not a diffeomorphism if our region ...
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128 views

Evaluating $\int_{0}^{\pi} \frac{1}{\sqrt{u^2+2u\cos x +1}} \mathrm{d}x$

Does the integral $$\int_0^{\pi} \frac{1}{\sqrt{u^2+2u\cos x +1}} \text d x \hspace{30pt} (u \le 1) $$ has a closed form? If it has, how do we evaluate it? I was solving a physics problem which I ...
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1answer
60 views

Challenging integration problem of an exponential function

While preparing a tutorial session for my students, I come across a very challenging integration, which is :$$ \int_{-\infty}^\infty e^{t^2 }\ \mathrm{d}t$$ I attempted many methods to solve it but ...
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92 views

How to solve this triple intergral $I=\iiint_{Q}\frac{1}{x^2+y^2}dv$?

$$ I \equiv \iiint_{Q}\frac{{\rm d}v}{x^2+y^2} $$ Which $Q$ is a solid bounded above by $z=4-x^2-y^2$ and below by the sphere $x^2+y^2+z^2=9$. I have tried with this multiple integration by using ...
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1answer
33 views

Show that $L(f)=\sup\{L(P,f):P \in \mathcal P_{c}\}$,where $c\in \left(a,b\right)$

Assume $\mathcal P_{[a,b]}$ is the set of all partitions of $[a,b]$ and $\mathcal P_{c}$ is the set of all partitions of $[a,b]$ containing $c$,where $c\in \left(a,b\right)$,if $f:I \to \mathbb R$ ...
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19 views

KL divergence between 2 joint probability distribution

Consider the following setting : $$Q(x,z)= q_{data}(x)q_{\phi}(z|x) \;\;\;\text{and} \;\;\; P(x,z)= p(z)p_{\theta}(x|z) = p_{\theta}(x)p_{\theta}(z|x)$$ I want to compute $KL(Q||P)$. Here is how I am ...
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1answer
38 views

Show that $m\le\left(\frac{1}{b-a}\int_{a}^{b}f^{2}\left(x\right)dx\right)^{\frac{1}{2}}\le M$

Assume $f$ is a real-valued function which is integrable over the interval $I=[a,b]$ and for every $x \in [a,b]$ we have that $0\le m \le f(x) \le M$,show the following inequality does hold: $$m\le\...
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8 views

Find isovalue such that a volume integral under the isosurface has a specific value

Lets say i have a continiously differentialbe function $f(x, y, z)$ that is nonnegative everywhere and has a finite volume integral over all space that yields 1, $$ \int^\infty_{-\infty }dx\int^\...
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1answer
45 views

$\int \frac{1}{x(x+1)…(x+n)}dx$ [duplicate]

I have not managed to evaluate the integral. I feel the idea is just split the integral into a sum of integrals $$\int \frac{1}{x(x+1)...(x+n)}dx = \int \left(\frac{A_1}{x} + \frac{A_2}{x+1} +... +\...
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1answer
34 views

Proof of generalized open & closed Newton-Cotes formulas

I am looking for proof of generalized open & closed Newton-Cotes formulas. I couldn't find any reference which properly proves both theorems. Most books just state the theorem and do not provide ...
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1answer
65 views

$I = \int_0^\infty y^2 [ (y^2 +1)/\sqrt{y^4 + 2 y^2 } - 1] dy$

I tried with contour integral: $$I = \frac{1}{2}\int_{-\infty}^{\infty } dz z^2\left(\dfrac{z^2 +1}{\sqrt{z^4 + 2 z^2 }} - 1\right)$$ The contour can be deformed into the upper half plane. But there ...
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33 views

Integral of a measurable not non negative function over N

I know that given a non-negative function $f: \mathbb{N} \rightarrow \mathbb{R}$, the integral over $\mathbb{N}$ with the counting measure is just the infinite sum. However, given the funcion $b(n) = \...
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73 views

Finding convolution of two functions

A common engineering notational convention is: wikipedia ${\displaystyle f(x)*g(x)\,:=\underbrace {\int_{0}^{x}f(\tau )g(x-\tau )\,d\tau } _{(f*g)(x)}.}$ I want to write the following expression as ...
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1answer
39 views

$\int_{0}^{\infty } e^{-t}\cdot t^{3}\cdot \sin(t) dt$ using Laplace transform

I am currently on this question: Use Laplace Transform to evaluate the following integral: $\int_{0}^{\infty } e^{-t}\cdot t^{3}\cdot \sin(t) dt$ What I did: let the $\int_{0}^{\infty } e^{-t } dt$ = ...
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1answer
52 views

Given a relation between $G$ and $g$ prove that $g(x)=x+1$.

I am having several difficulties in solving an exercise that has been put to me. The exercise says: If $g: (-1; \infty) \rightarrow \mathbb{R}^+$ is a function that can be antiderivated, let $G$ be ...
2
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0answers
83 views

Gauss Integration of $\sqrt{x}$

$w(x) = x^{1/2}$ $$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$$ $$\Pi_{2}(x^2 -p_{1}x+p_{2})$$ \begin{equation} \int_{0}^{1}x^{1/2} \cdot \Pi_{2}(x) dx = 0 \end{equation} \begin{equation}...
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36 views

Inverse Laplace Transform hint

I am away from Laplace transform for years, and now I have to solve $$\mathcal{L}^{-1} \left\{ s^{-\frac32}\sqrt{\frac{as+b}{cs+b}} \right\}$$a,b,c are real positive numbers.I can find inverse Laplace ...
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1answer
67 views

What does this definite integral theorem mean?

As I was solving a homework problem about definite integrals, I came across a theorem to help me solve the problem. Although I got the right answer to the problem, I do not really understand the ...
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1answer
27 views

On the calculation of curve integral

Let $\vec{n}$ be the out of unit normal vector on the curve $\varGamma$,and the $\varGamma:x^2+y^2=R^2$.define$$r=\sqrt{x^2+y^2}.$$Calculate$$\oint_{\varGamma}\dfrac{\partial\ln r}{\partial{\vec{n}}}\...
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21 views

Use the method of cylindrical shells to find the volume

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves x=sqrt(y), x=0 and y=8 about the x-axis.
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9 views

Implementing discrete derivative by integration over a set of data

I'm trying to implement Lanczos derivative $f'_L$ as an option to calculate over a set of data. Since i needed a discrete version of Lanczos derivative $f'_{DL}$, i rearranged the equation such as $$f'...
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1answer
23 views

Volume of a solid of revolution - Getting two different results.

The region between $y=x^2+1$ and $y=-x+3$ is rotated about the $x$-axis. I have to compute the volume. The intersection between these two curves are at $x=-2$ and $x=1$. At first, I thought about $V(x)...

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