Integral Definition and 1000 Threads

As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?

Hi colleagues This is a very very simple question I can show when $f$ is integrable and is even i.e. $f(-x)=f(x)$ then $\int_{-a}^{a} \,f(x)\,dx=2\int_{0}^{a} \,f(x)\,dx$ what about improper integrals of even functions, like the function ${x}^{2}\ln\left| x...

Problem Statement: The effective charge density of the electron cloud in a hydrogen atom in its quantum mechanical ground state turns out to be given by pnot(e^-(r/rnot)), where pnot is a negative constant (the clouds charge density at r=0) and rnot is a constant (rnot=0.025nm). Use gauss's law...

In a certain derivation, the author begins with $${g(-t)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(\omega)}e^{-i\omega t}d\omega$$ and then says he will replace ##t## with ##\omega## and ##\omega## with ##t##. He then writes $${g(-\omega)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(t)}e^{-it\omega...

how do I evaluate the double integral 2x + y dA over the region R bounded by y = 3x, 2X + y = 5, x = 0, and y = 0

I did the first part, it is part (b) that I'm having trouble understanding. For any ##x \lt b##, ##f(x)=0## and ##\int_0^x {f(t)} \, dt = 0## (since ##f## is 0 everywhere from 0 to ##b##), which turns the equation ##\int_0^x f(t) \, dt = (f(x))^2+C## into ##0=0+C##, which implies ##C=0##. But...

Suppose ##t \ge 0##. Let ##\displaystyle I(t) = \int_{-\infty}^{\infty}\frac{x \sin (tx)}{x^2+1}~\text{dx}##. Call this form 1. Note that we can also write the integral as $$ \begin{align*} I(t) &= \int_{-\infty}^{\infty}\frac{x \sin (tx)}{x^2+1}~\text{dx} \\ &=...

Hi I´d like a suggestion about a surface double integral. If I have a sphere x^2+y^2+z^2=4 is on the top of a cardioid r=1-cosθ. The problem is when I solve the integral I got a inverse sine when the answer is a natural logarithm (ln)

At first I was thinking about using the dirac delta function ##\delta(x-1)##, but then I recalled ##\delta \notin L_2[0,1]##. Any ideas? I'm thinking no such function exists.

I actually don't know how to proceed. I tried something like this The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$ where ##f'(x)=F(x)## However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$ where ##f'(x)=F(x)##. For the...

For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.

I want to understand very deeply the meaning of the work integral formula: \int m\frac{d\bar{v}}{dt}d\bar{l} It is not enough for me to know that it was defined in this way, I want to know why it was defined in this way. To start, what is the physical meaning of m\frac{d\bar{v}}{dt}d\bar{l}...

In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...

Evaluate the integral using the properties of definite integral and interpreting integrals as areas. ##\int_{-1}^2 (1-2x)dx## I need to see there are two areas and these are the same but one is under x-axis the other is above x-axis so the value of the integral is zero. To see this is...

Problem: Prove that for any $x \in R^n$ and any $0<p<\infty$ $\int_{S^{n-1}} \rvert \xi \cdot x \rvert^p d\sigma(\xi) = \rvert x \rvert^p \int_{S^{n-1}} \rvert \xi_1 \rvert^p d\sigma(\xi)$, where $\xi \cdot x = \xi_1 x_1 + ... + \xi_n x_n$ is the inner product in $R^n$. Some thinking... I...

Show that the value of ##\int_0^1\sqrt(1-cosx)dx## is less than or equal to ##\sqrt2## ##1\ge cos x\ge-1## The problem is a worked one but I am just confused by a simple thing. We integrate the function f ##\int_0^1\sqrt(1-cosx)dx in the interval [0,1] but I don't understand that what stands...

So i drew sketch. And I do not understand, how to write integral for calculation, which I should use, X or Y on limit? Is one of them right? First answer gives me 65,7 Second 383,4

I am able to solve the problem however if x was position and t was time how is this problem interpreted? I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.

Hi! $$\lim_{n \to \infty }\int_{0}^{n}\frac{dx}{1+n^{2}\cos^{2}x }$$ I found to solution on the internet but I didn't understood it 100%. First, it says that the function under integral has period $pi$.Why pi ? I know that cos function has period $2kpi$ Consequence: $\int_{0}^{k\pi...

Cant get my head around the order of the upper and lower bounds for this, Is it always the higher take away the lower?

Express the limit ##lim_{n\rightarrow\infty} \sum_{i=1}^n \frac2n\ (1+\frac {2i-1}{n})^\frac13## This is worked example but I would like to ask about the points I don't understand in the book. "We want to intepret the sum as a Riemann sum for ##f(x)=(1+x)^3## The factor ##\frac2n## suggests...

The integral is$$\int_0^4dz\iint xyz~dxdy$$Constricted to the quarter circular disk ##x^2+y^2=4## in the first quadrant. First I switched to polar coordinates and integrated the double integral by first writing it as:$$\int_0^4z~dz \int_0^\frac{\pi}2\int_0^2...

We often neglect the terms of a surface integral ##\int_v(\nabla•A)dv=\int_s(A•ds)## for ##s## to be very large or ##v## to be very large, What is actually the reason behind this to neglect??

Consider an integral of form $$\int_a^b dx f(x) g(x).$$ Is it possible to tell a numerical integrator to spit out the value of ##x \in [a,b]## that maximises the value of ##f(x)g(x)##? I'm mostly interested in incorporating this into some code I have for adaptive integrator gsl_qags in C++...

With respect to operations, I understand why an integral is multiplied by -1 when its limits reversed. But integral is geometrically an area so reversing the limits would not be able to change neither how large is the area nor the shape of the area. Would you please explain changing the limits...

Let ##V'## be the volume of dipole distribution and ##S'## be the boundary. The potential of a dipole distribution at a point ##P## is: ##\displaystyle\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{S'}\dfrac{\vec{M'}.\hat{n}}{r}dS'## If ##P\in V'## and ##P\in S'##, the...

I want to compute: $$\oint_{c} F \cdot dr$$ I have done the following: $$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$ I don't know what limits the surface integral will have. Actually, I am not sure what's the surface. May you shed some light...

Hi everyone, I am trying to find the definite integral of a function (see attached image) from 0 to infinity using Wolfram alpha. I'm just looking for some verification on if the integral actual is equal to exactly 1, or if there's some rounding errors going on. Thank you for your time :)

I split this to get \begin{equation} \int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx \end{equation} Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...

I'm having trouble understanding a specific line in my lecturers notes about the path integral approach to deriving the Klein Gordon propagator. I've attached the notes as an image to this post. In particular my main issue comes with (6.9). I can see that at some point he integrates over x to...

Let I = ##\int{\sqrt{\frac{cosx - cos^3x} {1-cos^3x}}}\,dx## I = ##\int{\sqrt{\frac{cosx(1 - cos^2x)} {1 - cos^3x}}}\,dx## I = ##\int{\sqrt{\frac {cosx} {1 - cos^3x }}}sinx\,dx## Substitute ##cosx = t## Therefore, ##sinx\,dx = -dt## So, I = ##\int{-\sqrt{\frac {t} {1 - t^3}}}\,dt## I'm stuck...

$$\int \frac {e^{1/x}} {x(x+1)^2} \, dx$$ I came across this indefinite integral when solving a second order differential equation using reduction of order. My CAS can solve it easy enough, but I was wondering what technique could be used to solve it by hand. I have tried some standard...

Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do...

I'm pondering something about properties of integrals. What can we say about the following limit? ##\lim_{t\to\infty} \int_t^{\infty } f(x) \, dx## On one hand, the 'gap' from the lower to upper integration limit diminishes, so that would suggest the limit is always 0. But what if f is an...

This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?

Homework Statement Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique. Homework Equations Starting equation integrated over t: int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1) Result...

Homework Statement Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y## Homework EquationsThe Attempt at a Solution [/B] I have already posted this question on MSE...

Homework Statement Homework Equations The Attempt at a Solution [/B] I understand how the integral is solved using cartesian coordinates. However, I wanted to try to solve it using polar coordinates: $$\int_0^{\pi/2} cos \theta \sqrt{1+r^2 cos^2 \theta}d...

The book on quantum mechanics that I was reading says: d<x>/dt = d/dt ∫∞-∞ |ψ(x,t)|2 dx =iħ/2m ∫∞-∞ x∂/∂x [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (1) =-∫∞-∞ [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (2) I want to know how to get from (1) to (2) The book says you use integration by part: ∫abfdg/dx dx = [fg]ab - ∫abdf/df dg dx I chose f...

How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...

Homework Statement 1) Calculate the density of states for a free particle in a three dimensional box of linear size L. 2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0## 3) Calculate the integral ##\int...

I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...

Homework Statement [/B] Hi in the first attachment I am stuck on the sign change argument used to get from line 2 to 3 , see below Homework Equationsabove The Attempt at a Solution [/B] Q1) please correct me if I'm wrong but : ##d^3 p \neq d\vec{p} ## since ##d^3 p = dp_x dp_y dp_z ## and...

Homework Statement Homework Equations So the question is asking to solve an integral and to use the answer of that integral to find an additional integral. With part a, I don't have much problem, but then I don't know how to apply the answer from it to part b. I know I should subsitute all...

I am reading "Inside Interesting Integrals" by Paul Nahin. Around pg. 59, he goes through a lengthy explanation of how to do the definite integral from 0 to infinity of ∫1/(x4+1)dx. However, he then simply writes down that this integral is equal to ∫x2/(x4+1)dx with the same limits. Now, it's...

Homework Statement hi my question is that say if rod of length l is 3m and is 90 degrees from horizontal pointing upwards with 2 kg weight on the end then there will be no torque produced because the force of gravity is acting parallel to the lever arm if the angle theta is 80 degrees then the...

Homework Statement Determine the series that is equal to the integral ##\int_0^1 x^2\cos(x^3)dx## Homework EquationsThe Attempt at a Solution So I didn't really know what I was doing but I did end up with the correct solution. What I did was to find a Taylor Series for the integrand, this...

Homework Statement $$\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) dxdzdy$$ Change the order of the integral to $$\iiint f(x,y,z) \, \mathrm{dydzdx}$$What I have done It is just about: From ##x=0## to ##x=\sqrt{4y+23}## From ##z=0## to ##z=4-y## From ##y=\frac{x^2-23}{4}## to...

Homework Statement Let D be the triangle with vetrices ##( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 )##. Evaluate the integral : $$\iint_D e^{\frac{y-x}{y+x}}$$ Homework EquationsThe Attempt at a Solution [/B] The answer to this problem is known (...

Homework Statement I attached an image of the multi-part problem on this post. I got correct answers to every question other than the last one. Homework EquationsThe Attempt at a Solution I believe the last part is a surface integral problem. F is given and I found n is previous parts of...