Integrals Definition and 1000 Threads

I'm going through a whole undergrad quantum book (Townsend) by myself. It has a chapter on path integral QM. He said in the intro that it can be skipped, but I was wondering if knowledge of this subject is immediately helpful when starting graduate level quantum. I start grad school in the...

Prove that \[\int_0^1 \frac{\log(1+x)\log(x)}{1-x}dx=\zeta(3)-\frac{\pi^2}{4}\log(2)\] \[\int_0^1 \frac{\log(1+x^2)}{1+x}dx=\frac{3}{4}\log^2(2) -\frac{\pi^2}{48}\]

Homework Statement Calculate the volume of the body that is bounded by the planes: x+y-z = 0 y-z = 0 y+z = 0 x+y+z = 2 Homework Equations The Attempt at a Solution I made a variable substitution u = y+z v = y-z w = x which gave me the new boundaries u+w = 2...

Homework Statement ∫1/(t*ln(t)) dt ∫1/(√(t)*[1-2*√(t)]) dt Homework Equations The Attempt at a Solution I used u-substitution for both. For the first equation, my u= ln t, and my final answer was ln|u| + C, or ln(ln(|t|) + C. For the second equation, my u= 1-2*√(t) and...

Is it safe to say when an integral has an infinite boundary \int_n^∞ a_{n} and the limit yields a finite number, then the integral is said to converge. And when a series has an upper limit of infinity \sum_n^{∞}a_{n} and the limit yields a finite number, then the series is said to diverge.

Huhh First of all I'm sorry if this is the wrong question ( didn't know if this was considered pre cal. I have a gut feeling it is :P) Homework Statement Let R be the region bounded by y=tan x, y=0 and x = Pi/4 Find the volume of the solid whose base is region R and whose cross-section...

Wasn't sure which section to put this q in. Just reading now that f(x,y) can represent the density of a semicircular wire and so if you take a line integral of some curve C and f(x,y) you can find the mass of the wire... makes sense. What I don't get is that if I then move the wire around the...

When I think line integral - I understand when I'm taking a line integral for a function f(x,y) which is in 3D space above a curve that the integral is this curtain type space, just like if you had a 2D function and you find the area under the curve, except now it's turned on its side and it's...

Learning about the Limit Comparison Test for Improper Integrals. I haven't gotten to any applications or actual problems yet. Just learning the theory so far, and have a question on the very beginning of it.Homework Statement f(x) ~ g(x) as x→a, then \frac{f(x)}{g(x)} = 1 (that is, f(x)...

Path Integrals-- Multivariable Calculus Hi all-- really stuck here, help would be greatly appreciated. :) 1. Evaluate ∫F
ds (over c), where F(x, y, z) = (y, 2x, y) and the path c is defined by the equation c(t) = (t, t^2, t^3); on [0, 1]: 2. Homework Equations L = sqrt(f'(t)^2 +...

Homework Statement Given 7 f(x) dx= 8 0 7 f(x) dx = −3 1 evaluate the following. 1 f(x) dx 0 Homework Equations n/a The Attempt at a Solution I'm a little confused on how to approach this problem. Do i use the additive interval property of integrals?

This thread will be dedicated to discuss the convergence of various definite integrals and infinite series , if you have any question to post , please don't hesitate , I hope someone make the thread sticky. 1- $$\int^{\infty}_0 \left(\frac{e^{-x}}{x}...

I know we have the following $$ \big | \int^{b}_{a} f(t) \, dt \big | \leq \int^{b}_{a} |f(t)|\, dt$$ 1- How to prove the inequality ,what are the conditions ? 2- Does it work for improper integrals ?

1. evaluate the following: \int^{1}_{0}\int^{1}_{0}xyex+y dydxThe Attempt at a Solution OK, so this should be pretty simple. But for some reason I am having trouble integrating the yex+y bit with respect to y. If I do it by parts I end up with iterations like this: \int^{1}_{0}xyex+y -...

Hi, I am reading through the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs and am having a bit of trouble with problem 3-12. The question is (all Planck constants are the reduced Planck constant and all integrals are from -infinity to infinity): The wavefunction for a...

Hello all, I started a thread exactly like this a few months ago. I really enjoy doing integrals, so if you could post some of the most challenging ones you know of, I would greatly appreciate it :smile: Jacob

How can I get the improper integral ## \int_0^{\infty}\frac{1}{x(1+x^2)}\,dx ## First thing I tried was separating the integral like this ## \int_0^{1}\frac{1}{x(1+x^2)}\,dx + \int_1^{\infty}\frac{1}{x(1+x^2)}\,dx## And then I tried with partial fractions but it didn't work

Homework Statement Evaluate ∫∫∫\sqrt{x^{2} + y^{2}} dA where R is the region bounded by the paraboloid y=x^2+z^2 and the plane y=4 Homework Equations I believe this is a problem where cylindrical coordinates would be useful 0 ≤ z ≤ \sqrt{4-x^2} 0 ≤ r ≤ 2 ( I think this is wrong). 0 ≤ θ ≤...

Homework Statement Check if the following integrals diverge: Int(1/(x^3-2x^2+x), from 0 to 1) and Int(1/(x^3+x^2-2x) Homework Equations Ratio-test(not sure if that's the name)The Attempt at a Solution I have solved the problem and found that both integrals diverge at x=0. I just want to check...

Homework Statement A cylinder has a diameter of 2 inches. One end is cut perpendicular to the side of the cylinder and the other side is cut at an angle of 40 degrees to the side. The length at the longest point is 10 inches. Find the volume of the sample. I believe this is what it would...

Homework Statement Integrate some area C of (xe^y)ds where C is the arc of the curve x=e^y Homework Equations What is the indeffinite integral and why is it that? Answer is (1/3)e^3y + C The Attempt at a Solution Integral of (xe^y)((e^y)^2 + 1)^(1/2) = Integral of (e^2y)(e^2y...

1. Homework Statement [/b] this problem is on page 267 of Advanced calculus of several variables by Edwards, I just can't seem to get a handle on it: Let aA be a contented set in the right half of the xz plane ,x>0. Define $$\hat{x}$$, the x-coordinates of the centroid of A, by...

Hi everyone. Just for fun I thought we could post some of the more interesting ways we know of to evaluate integrals :) For starters, to evaluate \displaystyle \begin{align*} \int{\arctan{(x)}\,dx} \end{align*}, first we consider the integral \displaystyle \begin{align*} \int{\frac{2x}{1 +...

I'm working on simple game and am working on a leveling system, using a function to get experience needed. I am using area under a function above y=0. The first problem, I can't figure out a simple number. f(x) = x2/5 dx Then, looking for area, I'm unsure about a really simple thing. Getting...

Hi everyone I have these 2 integrate that I can't solve, I have tried them with mathematica and wolfram, but they can't find an answer, maybe someone have an idea on how I could tackle these 2 bad boy! The first one is $$\int{ \sqrt{ \frac{1+( \frac{1}{10}+ \frac{s}{25})^2}{ \frac {s}{10}+...

Hello MHB, I would like to have tips how to solve the x limits for this problem [FONT=Times], [FONT=Times]there Regards,

Hello MHB, I got as homework to solve this problem and get recommend to solve it with polar but I have not really work with polar but we have had lecture about it and I have done some research. This is the problem and what I understand $$\int\int_Dx^3y^2\ln(x^2+y^2)$$, $$4\leq x^2+y^2\leq 25$$...

Homework Statement Find the Volume of the solid eclose by y=x^{2}+z^{2} and y=8-x^{2}-z^{2} The Attempt at a Solution Well know they're both elliptic paraboloids except one is flipped on the xz-plane and moved up 8 units. Knowing this, i equated the two equations and got...

Hello MHB, Exemple 3: "Evaluate $$\int\int_D xy dA$$, where D is the region bounded by the line $$y=x-1$$ and parabola $$y^2=2x+6$$" They say I region is more complicated (the x) so we choose y. so if we equal them we get $$x_1=-1$$ and $$x_2=5$$ then as they said it's more complicated if we...

Hello MHB, I wanted to 'challange' myself with solve a problem with midpoint and rule and the double integral f over the rectangle R. This is a problem from midpoint. "Use the Midpoint Rule m=n=2 to estimate the value of the integrab $$\int\int_r(x-3y^2)dA$$, where $$R= {(x,y)| 0\leq x \leq 2, 1...

We are already introduced to finding the value of definite integral by the anti-derivative approach \int_{a}^{b}f(x) dx = F(b) - F(a) In this approach we find the anti-derivative F(x) of f(x) and then subtract F(a) from F(b) to get the value of the definite integral Reimann Sums...

Hi, I've recently taken a Calculus 1 (Differential Calculus) course and I've been looking ahead to see what sort of material is covered in the Calculus 2 (Integral Calculus) course. I am wondering about the relationship between derivatives and integrals. From what I understand, an integral...

Is it possible to claculate ∫[cot(x)]dx from x=0 to x=\piwhere [.] represents floor functon or the greatest integer function?? it seems impossible to me but can we use the properties of definite integrals to somehow evaluate the area?

When evaluating an improper integral and i get infinity minus infitiy when taking the limit. in what case do the areas cancel?

I'm using Green's Functions for heat conduction problems, and I'm trying to solve the following integral: Homework Statement http://img28.imageshack.us/img28/4923/026307b169b04faa8364086.png Where: http://img820.imageshack.us/img820/3742/6332938c445f4b9e8da8ba5.png Homework...

In trying to solve \int^{\infty}_{-\infty} x + \frac{1}{x} dx could it be split up and solved using the Cauchy Principle Value theorem and a contour integral along a semi-circle. Thus; PV\int^{\infty}_{-\infty}x dx =0 +\int \frac{1}{x} dx = \int^{\pi}_{0} i d\theta Is this valid reasoning?

1) Show that for $\alpha$ not an integer multiple of $\pi$, $\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)} $.2) Show that for $s,\lambda >0$ and $0 \le...

Homework Statement This is a book problem, as follows: Find the volume between the cone x = \sqrt{y^{2}+x^{2}} and the sphere x^{2}+y^{2}+z^{2} = 4 Homework Equations spherical coordinates: p^{2}=x^{2}+y^{2}+z^{2} \phi = angle from Z axis (as I understand it) \theta = angle from x or...

Homework Statement When integrating a function of the form: \displaystyle\int_c^d { (sinx)^{a} * (cosx)^{b}} Is this a correct simplification of the rules to evaluate: 1. if exponent on sin or cos is odd, and the other is even, separate out one of the odd's and use an identity on the...

Homework Statement Find the volume obtained by rotating the following curves bounded by: y=sinx y=0 pi/2≤ x ≤ pi Homework Equations I know I have to use the cylindrial disk method so ∫2pi(x)f(x)dx. The Attempt at a Solution I did the following ∫(pi/2 to pi) 2pi (x) sinxdx...

An area A in the xy-plane is defined by the y-axis and by the parabola with the equation x=6-y^2. Furthermore a surface S is given by that part of the graph for the function h(x,y)=6-x-y^2 that satisfies x>=0 and z>=0. I have to parametrisize A and S. Could this be a...

Homework Statement ∫e-Sxsin(ax) dx, S and A are constants, upper limit is ∞ lower is 0 Homework Equations ∫ u dv = uv - ∫ vdu The Attempt at a Solution After integrating by parts twice I got: (S2)/S(S2+a2) lim c→∞ [-sin(ax)e-Sx + acos(ax)e-Sx] |^{C}_{0} Okay, now how on Earth do I take...

Suppose we have the double integral of a function f(x,y) with domain of integration being some rectangular region in the 1st quadrant: 0≤a≤x≤b, 0≤c≤y≤d. Would the following transformation generally be acceptable? (I've quickly tried it out several times with arbitrary integrands and domains...

Homework Statement 25-2-EX9 The time rate of change of the displacement (velocity) of a robot arm is ds/dt = 8t/(t^2 + 4)^2. Find the expression for the displacement as a function of time if s = -1 m when t = 0 s. 26-5-9 Find the moment of inertia of a plate covering the first-quadrant region...

I did a proof a few days ago (for the sake of enjoyment) and my teacher thought it was interesting, though he seemed unsure of my result. Consider a set of n distinct objects, P. If n \in \mathbb{Z}_+ \cup \left\{0\right\}, then the cardinality, q, of the set of all derangements of P is...

{\frac{∂(xy)}{∂x}=x} Going backwards. If we took, ∫x dy we get xy+f(x) Now, the only way that ∫x dy is a valid operation, is if we know that we came from a partial derivative. Why, when taking a partial...

I suppose you all know the substitution formula for integrals. Well sometimes it seems to me you use substitutions which just don't fit directly into that formula. For instance for the integral of 1/(1+x^2) you substitute x=tan(u). Why is it suddenly allowed to assume that x can be...

Hi I'm currently studying Electromagnetism, and we keep coming across this symbol: \oint A closed line integral, something I have never really been able to understand. If a normal integral works like this: http://imageshack.us/a/img109/3732/standardintegral.png where f(x) is the "height"...

Homework Statement Given \int^{\sqrt{6}}_{0}\int^{x}_{-x}dydx, convert to ploar coordinates and evaluate. Homework Equations We know that x=rcos\theta and y=rsin\theta and r =x^2+y^2 The Attempt at a Solution First, I defined the region of the original integral: R = 0...

Homework Statement A fluid as velocity field F(x, y, z) = (xy, yz, xz). Let C denote the unit circle in the xy-plane. Compute the circulation, and interpret your answer.Homework Equations The Attempt at a Solution Since the unit circle is a closed loop, I assumed that ∫ F * dr = 0 (the ∫ symbol...