Integrals Definition and 1000 Threads

Homework Statement a) sketch the region in the first octant bounded by the elliptic cylinder 2x^2+y^2=1 and the plane y+z=1. b) find the volume of this solid by triple integration. Homework EquationsThe Attempt at a Solution I have already sketched the elliptic cylinder and the plane. my...

Homework Statement I am having trouble setting up the bounds on the following two integrals: (a) The region E bounded by the paraboloid y=x2+z2 and the plane y=4. (b) The region bounded by the cylinder x2+y2=1, z=4, and the paraboloid z=1-x2-y2. Homework EquationsThe Attempt at a Solution I...

I am having some trouble with finding the boundaries for the first part of the problem (dz dy dx), I should be able to figure out the second part on my own. The problem is: Set up the triple iterated integrals (using dz dy dx and d θ dr dz) to find ∫∫∫E \sqrt{x^2+y^2} dV where E is the part of...

How does one prove the following relation? \int_{a}^{b}f(x)dx= \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx Initially, I attempted to do this by writing the definite integral as the limit of a Riemann sum, i.e. \int_{a}^{b}f(x)dx=...

when taking double or triple integrals, do you get the same solution no matter which variable you integrate with respect to first?

Homework Statement Suppose we have the function ##f : I \rightarrow \mathbb{C}##, where ##I## is some interval of ##\mathbb{R}## the functions can be written as ##f(t) = u_1(t) + i v(t)##. Furthermore, suppose this function is integral over the interval ##a \le t \le b##, which can be found by...

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 >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 exam I saw...

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 An important problem in thermodynamics is to find the work done by an Ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region R enclosed by two isothermal curves xy = a...

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 that they...

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...