Integrals Definition and 1000 Threads

Homework Statement Homework Equations ∫∫D F((r(u,v))⋅(ru x rv) dA The Attempt at a Solution [/B] I got stuck after finding the above, at where the double integrals are. :( May I know how do I find the limits of this? (I always have trouble finding the limits to sub into the integrals...

Two (supposedly) trival questions in Schwartz's QFT notes. The notes can be found http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf. 1. page 155, equation 15.2, how does the integrand reduce to k dk? I would guess that there must be some logarithm, but k dk? 2. page 172...

Homework Statement Determine whether or not f(x,y) is a conservative vector field. f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) > If F is a conservative fector field find F = gradient of f Homework Equations N/A The Attempt at a Solution Fx = -3e^(-3x)(-3)cos(-3y) Fy =...

Evaluation of $$\displaystyle \int_{0}^{\pi}\lfloor \cot x \rfloor dx$$ and $$\displaystyle \int_{0}^{\pi}\lfloor \cos x \rfloor dx\;,$$ where $$\lfloor x \rfloor $$ denote Floor function of $$x$$

< Mentor Note -- thread moved from General Math to the Homework Help forums >[/color]Hi all, Calc II finals is 4-5 weeks away...We're on Taylor Series right now, but I wanted to get started early on studying for the final. I have a few questions that are confusing me that I took from a final...

Homework Statement The problem is given in the attached file. Homework Equations Divergence theorem, flux / surface integral The Attempt at a Solution [/B] As you can see I got the question correct using Divergence theorem. But I wanted to make sure that I could arrive at the same answer...

Homework Statement \int_{-\infty}^{\infty} \frac{\sin(x)}{x} using Complex Analysis Homework Equations Contour analysis on \int_{-\infty}^{\infty} \frac{\sin(x)}{x} The Attempt at a Solution Hello, I am completely new to contour integration. I would really appreciate it if someone can walk...

Homework Statement Evaluate ∫∫ F⋅dS, where F = yi+x2j+z2k and S is the portion of the plane 3x+2y+z = 6 in the first octant. The orientation of S is given by the upward normal vector. Homework Equations ∫∫S F⋅dS = ∫∫D F(r(u,v))⋅||ru x rv|| dA, dA=dudv The Attempt at a Solution [/B] Since...

Homework Statement Many places I have seen when solving integrals you change a lot of it into sums. http://math.stackexchange.com/questions/1005976/finding-int-0-pi-2-dfrac-tan-x1m2-tan2x-mathrmdx/1006076#1006076 Is just an example. So in general, how do you solve integrals (CLOSED FORM) by...

Homework Statement ∫y1(x)^2dx from - to + infinity=1 and ∫y2(x)^2dx from - to + infinity=1 Homework Equations None that I know of. The Attempt at a Solution I evaluated the integrals and got that c1 is equal to c2 but I think that's wrong.

Hello, Suppose we have: $$\begin{align} \sum_{n=1}^{\infty}\frac{1}{9n^2 + 3n - 2} &=\frac{1}{3}\sum_{n=1}^{\infty}\left(\frac{1}{3n - 1}-\frac{1}{3n + 2}\right)\\\\ &=\frac{1}{3}\sum_{n=1}^{\infty}\int_0^1\left(x^{3n-2}-x^{3n+1}\right){\rm d}x\\\\...

So I am kind of confused about the role of force when calculating work. Specifically, when defining work using a line integral. There is a paragraph in my calculus book that is really throwing me off and its really bugging me so much I can't continue reading unless I fully understand what's...

Hello, I have began my journey on infinite sums, which are very interesting. Here is the issue: I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle...

Be a vector field \vec{F}=(f_1,f_2,f_3) and \omega^k_{\vec{F}} the k-form associated with it , i know if i do \int \omega^1_{\vec{F}} is the same of a line integral and \int \omega^2_{\vec{F}} i obtain the same result of \int \int_S \vec{F}\cdot d\vec{S}, which is the flux of a vector field in a...

Need help on exercise 2 from the linked image , left first in so you guys could see the Γ(χ) function any help is appreciated , thanks in advance!

I read that the improper Riemann integral ##\int_0^1 \frac{1}{x}\sin\frac{1}{x}dx## converges and that ##\int_0^1 |\frac{1}{x}\sin\frac{1}{x}|dx## does not. I have tried comparison criteria for ##\int_0^1 |\frac{1}{x}\sin\frac{1}{x}|dx##, but I cannot find a function ##f## with a divergent...

Homework Statement Prove the following property: If m <= f(x,y) <= M \hspace{2 mm} \forall (x,y) \in D, then: mA(D) <= \int\int f(x,y)\,dA <= MA(D) Homework Equations I use a few other known properties in the proof (see below) The Attempt at a Solution First, I should state that this problem...

Homework Statement Evaluate the following double integral: V = \int\int \frac{3y}{6x^{5}+1} \,dA D = [(x,y) \hspace{1 mm}|\hspace{1 mm} 0<=x<=1 \hspace{5 mm} 0<=y<=x^2] Homework EquationsThe Attempt at a Solution V = \int_{0}^{1} \int_{0}^{x^2} \frac{3y}{6x^{5}+1}\,dy\,dx =...

I can't compute the integral: \int \frac{\arccos(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}\frac{x-a}/{(\sqrt{(x-1)^2+y^2})^3 dxdy on an unit circle: r < 1. for const: a = 0.01, 0.02, ect. up to 1 or 2. I used a polar coordinates, but the values jump dramatically in some places (around the 'a' values)...

Homework Statement integrate (-3csc(theta))/(1+cos(theta)) Homework Equations i'm not sure The Attempt at a Solution i tried using u sub. but i got nowhere. U=1+costheta Du=-sintheta

Problem Show: \int_0^\infty \frac{cos(mx)}{4x^4+5x^2+1} dx= \frac{\pi}{6}(2e^{(-m/2)}-e^{-m}) for m>0 The attempt at a solution The general idea seems to be to replace cos(mx) with ##e^{imz}## and then use contour integration and residue theory to solve the integral. Let ##f(z) =...

Homework Statement Find the exact length of the curve: y= 1/4 x2-1/2 ln(x) where 1<=x<=2 Homework Equations Using the Length formula (Leibniz) given in my book, L=Int[a,b] sqrt(1+(dy/dx)2) I found derivative of f to be (x2-1)/2x does that look correct? The Attempt at a Solution I found f'...

from 0 to π/2 ∫sin5θ cos5θ dθ I have been trying to solve the above for quite some time now yet can't see what I am doing wrong. I break it down using double angle formulas into: ∫ 1/25 sin5(2θ) dθ 1/32 ∫sin4(2θ) * sin(2θ) dθ 1/32 ∫(1-cos2(2θ))2 * sin(2θ) dθ With this I can make u = cos(2θ)...

Stressed first year university student here, fresh out of high school. I took physics in both grade 11 and 12, and thought I had a pretty good grasp on it; that is until this week. Introduction to derivatives and integrals to get from x(t) to v(t) to a(t) and vice-versa. I have a pretty good...

are nonelementary integrals implicit functions? ie, when we do implicit differentation, we get an explicit function. What if i go the opposite way, and integrate an explicit function to get an implicit antiderivative?

Homework Statement We know that F(x) = \int^{x}_{0}e^{e^{t}} dt is a continuous function by FTC1, though it is not an elementary function. The Functions \int\frac{e^{x}}{x}dx and \int\frac{1}{lnx}dx are not elementary funtions either but they can be expressed in terms of F. a)...

True or False? Let a and b be real numbers, with a < b, and f a continuous function on the interval [a, b]. a) If a=b then \int^{b}_{a} f(x)dx = 0 My answer: This is TRUE, because while this integral would have a height, it would NOT have a width and area being l*w will result in 0. b) If a...

Homework Statement Evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. f(x) = \frac{1}{x^2+1} at the point (1,1/2) Homework Equations The Attempt at a Solution So far I...

Also, how would you do $\int \sqrt{x^2-4}$? $$d(x^2-4)^{3/2}=3x\sqrt{x^2-4} \,dx$$ $$\int \frac{\sqrt{x^2-4}}{3x}d(x^2-4)^{3/2}$$ Not sure how partial integration will be useful here. What standard integrals do you see?

Hi. I'm off to solve this integral and I'm not seeing how \int dx Hm(x)Hm(x)e^{-2x^2} Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here. Using Hm(x)=(-1)^m...

Dear all, I am self studying GR and stuck on problem (23) on page 108/109. I am trying to do all of them. First I will start with (a) so you guys can breath while laughing at my attempts at (b) and (c) :blushing: (a) Attempt The tensor in the equation is bounded in the d^{3}x region. Outside...

Hi So let's have ∫(2x)/(4x^(2)+2) dx Without factorising the 2 from the denominator, I integrate and I get 1/4*ln(4x^(2)+2)+c which makes sense as when I differentiate it I get the original derivative. BUT when I factor the 2 from the denominator I have 2x/[2(2x^(2)+1)]...

I have read that even the simplest integrals (like y=x2) might need some correction if we want to reach an extreme precision. Is that really so? Can you explain why or give me some useful links? Thanks

Can anyone suggest me a good reference for path integrals (QFT), apart from peskin.

Ok for the purpose of this question let's stick to the flux integral: The general formula is ∫∫s (E-vector)*(dS-vector)=Flux where * stands for the dot-product. Now, I like it when my integrals make sense, and to do that I usually think of the Riemann Sum which might represent my integral...

Suppose I want an expectation value of a harmonic oscillator wavefunction, then in what way will I write the Hermite polynomial of nth degree into the integral? I have a link of the representation, but don't know what to do with them? http://dlmf.nist.gov/18.3

Homework Statement ##\nabla{F} = <2xyze^{x^2},ze^{x^2},ye^{x^2}## if f(0,0,0) = 5 find f(1,1,2)Homework Equations The Attempt at a Solution my book doesn't have a good example of a problem like this, am I looking for a potential? ##<\frac{\partial}{\partial x},\frac{\partial}{\partial...

From textbooks, I usually see that when there is an integral like this: $$\int_{-\infty}^{+\infty} f(x)\,dx$$, they generally split it two, usually by 0. $$\int_{-\infty}^{0} f(x)\,dx + \int_{0}^{\infty} f(x) \,dx$$ They do the same for points of discontinuity, but if you notice, the number...

Homework Statement Set up the double integral over the region ##y=x+3; y=x^2+1## Homework Equations The Attempt at a Solution finding the intersections you get the double integral ##\int_{1}^{5}\int_{-1}^{2}dxdy =12 ## but why is that not the same as...

Homework Statement I am given W = \{ (x,z,z)| \frac{1}{2} \le z \le 1; x^2 + y^2 +z^2 \le 1\} they want the iterated integrals to be of the form \iiint_W dzdydx The Attempt at a Solution so I know z=1/2 will give me the larger bound for x x^2 + y^2 + (1/2)^2 =1...

Definition/Summary This article is a list of standard integrals, i.e. the integrals which are commonly used while evaluating problems and as such, are taken for granted. This is a reference article, and can be used to look up the various integrals which might help while solving problems...

Homework Statement \int_{1}^{4}\int_{1}^{\sqrt{x}}(x^2+y^2)dydx The Attempt at a Solution I drew the region, I tried \int_{1}^{2}\int_{1}^{y^2}(x^2+y^2)dxdy but it doesn't seem to work. when the order is changed 1 \le y \le 2 and \sqrt{x} = y \rightarrow...

I think I may have found an error in the text I'm reading. Here's a quote: ... + \int_0^{\infty}x^rf_1(x)sin(2\pi logx)dx. However, the transformation y=-logx-r shows that this last integral is that of an odd function over (-∞,∞) and hence equal to 0 for r=0,1,... By the way, the author means...

Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...

Homework Statement Let F = <z,x,y>. The plane D1: z = 2x +2y-1 and the paraboloid D2: z = x^2 + y^2 intersect in a closed curve. Stoke's Theorem implies that the surface integrals of the of either surface is equal since they share a boundary (provided that the orientations match)...

how to solve triple integrals in cylindrical, spherical and rectangular coordinates ..easy ways

I know the formula for a change of variables in a double integral using Jacobians. $$ \iint_{S}\,dx\,dy = \iint_{S'}\left\lvert J(u,v) \right\rvert\,du\,dv $$ where ## S' ## is the preimage of ## S ## under the mapping $$ x = f(u,v),~ y = g(u,v) $$ and ## J(u,v) ## is the Jacobian of the mapping...

Is there a formal relation that links \int yxdx OR \int_{a}^{b}yxdx with \int xydy OR \int_{a}^{b}xydy where y=f(x) over the interval x\in\left[a,b\right].

How does one prove the following: \int^{c}_{a} f\left(x\right)dx = \int^{b}_{a} f\left(x\right)dx +\int^{c}_{b} f\left(x\right)dx where f\left(x\right) is continuous in the interval x\in \left[a, b\right], and differentiable on x\in \left(a, b\right). My approach was the following...

Homework Statement ∫∫[ye^(-xy)]dA R=[0,2]×[0,3] evaluate the integral. Homework Equations The Attempt at a Solution So I started with some algebra changing the integral to ∫(e^-x)[∫ye^-ydy]dx I evaluated the y portion first because its more difficult to deal with and wanted to...