Integrals Definition and 1000 Threads

Homework Statement Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is bounded by the parabola x=y2 and the line y = x - 2; ρ(x, y)=3Homework Equations m=\int\intD ρ(x, y) dA The Attempt at a Solution Basically I just need help...

Homework Statement The portion of the paraboloid 2z=x^2+y^2 that is inside the cylinder x^2+y^2=8 The Attempt at a Solution my attempt was that i would turn this into polar coordinates and solve that integral but is it right? I came up with...

I can't seem to understand why, if there are s generalized coordinates, there end up being only 2s-1 integrals of the motion. The solutions of Lagrange's equation will have 2s constants. Why couldn't one simply solve the 2s equations for the solutions q_i and dq_i/dt for the 2s constants...

In a recent homework assignment, I was asked to prodive a definition for ∫f(x) in the Region D, provided there was a discontinuity somewhere in the region. To define the integral, we merely removed a sphere centered on the discontinuity of radius δ>0 and found the limit of the integral as δ→0...

[PLAIN]http://img31.imageshack.us/img31/9004/screenshot20111117at720.png Proofs always get to me for some reason. It's like other problems I can do, but when it comes to proofs I don't know what to put. Can anyone show me steps? Thank you

I'm having trouble figuring out how to find what "r" is. I know r is the radius, but how do I go about finding it? Like what do I look for in a particular problem?

Homework Statement ∫xe^(-2x)dx from x = 0 to ∞ Homework Equations -xe^(-2x)/2 - e^(-2x)/4 + C The Attempt at a Solution lim b→∞ -x/2e^(2b) - 1/4e^(2b) = 0 wolfram alpha says its 1/4 and I do not know why (it does not show steps) Can you help me?

I'm looking for some tricky/difficult integrals within the scope of calc I and II that I can play around with. Most of the integrals in my books (Stewart and Spivak) are fairly straight forward, and the only real practice I get is in "rigor". I can't really make up my own problems either...

Homework Statement There is a circle of equation x^2+y^2=1 and a vector field F (x; y) =< y + .5x, x + .3y >. Imagine the field zoomed in extremely close at (0,1), to the point where it looks like a constant field of <-1,.3>. Calculate the work from say (0,1) to (-.001, 1). The constant field...

I am just trying to get the conceptual basics in my head. Can I think of things this way... If you are taking the integral of a function f(z) along a curve γ in a region A. If the curve is closed and f(z) is analytic on the entire curve as well as everywhere inside the curve, then the...

I've been looking for some good resources on integrals in four-space (SR and GR), and hope someone can suggest some! I'm not too interested in abstract mathematical formalisms to the extent of pure math though, I must keep in mind that this is all to do with physics (at least for me!). I know...

The solid enclosed by the cylinder x^2 + y^2 = 9 and the planes y + z = 5 and z=1. The biggest part for me (usually) is just being able to find my limits of integration for these problems (any suggestions about that would also be greatly appreciated). I think I found the correct limits for...

Homework Statement Let A = \{(x, y, z) \in \mathbb{R}^n : 0 \lt x \leq 1, 0 \lt y \leq 1 - x^2, 0 \lt z \leq x^2 + y\}. Define f : A \rightarrow \mathbb{R} by f(x, y, z) = y for each (x, y, z) \in A. Accept that Fubini's theorem is applicable here. Find \int_A f. Homework Equations Fubini's...

Hi! Here's a question I am working on: Double integral of arctan(y/x). where R: 1≤x2+y2≤4, 0≤y≤x. I have the bounds for r as 1 to 2, but for θ I don't know if I should use ∏/4 to ∏/2 or 0 to ∏/2. How do I know which one? The integration is easy, but I need help with the bounds...

Homework Statement The domain D is the intersection of two disks x^2 +y^2 = 1 and x^2 + (y-1)^2 =1 use polar coordinates to find the double integral ∫∫(x)dA Homework Equations x = rcosθ y = rsinθ r^2 = x^2 + y^2 The Attempt at a Solution I have drawn the circles...

Say f is a non-negative, integrable function over a measurable set E. Suppose \int_{E_k} f\; dm \leq \epsilon for each positive integer k, where E_k = E \cap [-k,k] Then, why is it true that \int_E f\; dm \leq \epsilon \quad ? I know that \bigcup_k E_k = E and intuitively it seems...

I would like to know how the following integral for the error function gets derived (found by following the link): http://www.wolframalpha.com/input/?i=integrate+exp%28-x%5E2%29+from+x0+to+inf Note: this is not a homework question, merely a query.

Homework Statement Evaluate the integral, where E is the solid in the first octant that lies beneath the paraboloid z = 9 - x2 - y2. ∫∫∫(2(x^3+xy^2))dV Homework Equations x=rcosθ y=rsinθ x^2+y^2=r^2 The Attempt at a Solution θ=0 to 2π, r=0 to 3, z=0 to (9-r^2)...

hi experts as far I know the stokes theorem relates surface integral to line integral - but i am confuse how surface integral if represent area gets equal to length as represented by line integral.

Homework Statement With the time evolution time operator, where there is time dependent hamiltonian, show the new form of the feynman propagator between two states. Consider the Weyl Integral. 2. Equations from \newcommand{\mean}[1]{{<\!\!{#1}\!\!>}}...

Just got into improper integrals, in my Calculus 2 class. We're looking to see if the integral converges or diverges. Homework Statement The integral given: ∫(dt/(t+1)^2) on the interval from -1 to 5Homework Equations uhhh... The Attempt at a Solution Took the limit as "a" goes to -1. Did a...

I've been having some trouble with a maths problem and I hoped someone might be able to help. We don't seem to have been taught most of what we need to do this, I understand Riemann integrals but what we've been taught and what they're asking for is just different. I could do with a...

Hi, I applied Gauss-lobatto method to solve elliptic integral in my problem. one of my parameters(distance) is infinity. in my problem the answer of elliptic integral at large distance should be zero but I can't get this result numerically. when distance goes to infinity, my program stopped...

I've been reading Mechanics of Landau Lifgarbagez. They state that "not all integrals of motion are of equal importance", and that "there are some whose constancy is of profound importance"...these ones are conserved for the motion. What confuses me is that I thought that's what an integral of...

Homework Statement What I want to show is this: ∫|x+y| ≤ ∫|x| + ∫|y| Homework Equations |x+y| ≤ |x| + |y| The Attempt at a Solution So I thought if I used the triangle inequality I could get to something along the lines of: Lets g belong to the real numbers ∫|x+y| =...

Homework Statement Solve integrals \int^0_{-\infty}e^{(a-ik)x}dx \int^{\infty}_{0}e^{-(a+ik)x}dxHomework Equations \int e^x=e^x+C The Attempt at a Solution My troble is with imaginary unit i \int^0_{-\infty}e^{(a-ik)x}dx=\frac{e^{(a-ik)x}}{a-ik}|^0_{-\infty}...

Homework Statement Demonstrate that \int_{-\gamma} f(z)|dz|=\int_{\gamma} f(z)|dz| where \gamma is a piecewise smooth path and f is a function that is continuous on |\gamma|. Homework Equations The Attempt at a Solution This proof seems like it should be very simple, but I am...

Homework Statement I have a problem as follows: Let \gamma=\beta+[e^2\pi,1] where \beta is given by \beta(t)=e^{t+it} for 0\leq 2 \leq \pi. Evaluate \int_\gamma z^{-1} dz . Homework Equations The Attempt at a Solution I know that I need to parameterize the path and I have...

Homework Statement Show for the z-component of the curl of a vector function that the integral off this over an infinitesimal rectangle in the x-y plane is equal to the contour integral of the original vector function around the perimeter. Homework Equations Short of explaining how to...

Hey, This is just a small question about Cauchys theorem. If there is a function f(z) such that int f(z)dz = 0 can you conclude f is analytic in and on the region of integration? What I mean is can you work the theorem in reverse? For example if the above is true over a region C...

Its been a while since I took calculus so I'm confused as how to solve this. I've gotten my equation simplified as far as BdT=KdP and I'm supposed to solve for dP I do it and end up with B(T2-T1) = K(P2-P1) but this is giving me the wrong answer when I put the values in... What...

Hi, I understand that from my EM class there exist a surface integral which is actually a way of summing infinitesimally small surface elements ds. But then I ran into some theorems on internet and I saw the denotation of double integral, over a surface S. And they called that a surface...

Homework Statement If we have volume integral of a Gaussian function, in phase space for example. F= \int^{\infty}_{-\infty} e^{-aq} d^{3}q Now, I think the the answer would be the standard answer for a Gaussian integral cubed wouldn't it? F=\left(\frac{\pi}{a}\right) ^{3/2} I...

Homework Statement A sleeve of mass m is constrained to move without friction along the x-axis. The sleeve is connected to the point (0, 2) on the y-axis by a spring as shown in the diagram below. Assume that Hooke’s “Law” is a good approximation for the restoring force exerted by the...

Homework Statement A cylindrical coffee cup (8 cm in diameter and 10 cm tall) is filled to the brim with coffee. Neglecting the weight of the cup, determine the torque at the handle (2 cm from edge of cup 5 cm up from bottom of cup). The easy way would be to just use the center of mass of the...

Hi all, can you help me to compute these integrals? \int \frac{x \sqrt{a x+b+x^2}}{d^2+x^2} \, dx \int \frac{x}{\left(d^2+x^2\right) \sqrt{a x+b+x^2}} \, dx a,b and d are real and positive. I tried with Mathematica, but the results involve logarithmic functions with complex...

Hi all, I have a family of functions defined by integrals and indexed by n, e.g f[x_,n_]=\int dy e^{ixy}y^n Is it possible to evaluate the integrals corresponding to different particular values of n in such a way that mathematica "remembers" that say f[x,4]= some function g[x]? An additional...

Homework Statement can anyone explain me about Integrals and derivatives in brief Homework Equations The Attempt at a Solution

Homework Statement There are two integrals that I don't know how to solve. I'm fairly certain the preceding work that led me to these integrals is correct. Homework Equations 1. ∫sin (x) / (csc (x) + cot (x)) 2. ∫x^(2) / (1 - x) The Attempt at a Solution U substitution didn't...

Homework Statement integral 1/x^(2/3)dx from -1 to 1 Homework Equations The Attempt at a Solution so i split it up into two integrals, one with limits going from -1 to b and the other with limits going from c to 1, and taking the limits as b and c go to 0 i know my antiderivative...

Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...

Hi I have not studied calculus for a while and I am just seeking some clarification on the following two problems I have attempted to solve. PROBLEM 1 dy/dx = y(x^3 - √x) I have separated the variables as follows: Rewrote equation as dy/dx = y(x^3 - x^1/2) Divided both sides by...

Hi I have a little problem with the integrals of the following functions. integral from 0to 2pi ∫sin^2(nθ+ψ)dθ=∫cos^2(nθ+ψ)dθ=pi ψ=the phase angle.They occur in the theory of vibrations. Is it appropriate to set ψ=0 Thank you

Homework Statement \int cos^6(x) Homework Equations 1 = sin^2(x) + cos^2(x) The Attempt at a Solution \int cos(x) * cos^5(x) \int cos(x) * (cos^2(x))^3 \int cos(x) * (1-sin^2(x))^3 This is where I got lost, we just started this topic and I have a lot of homework to do...

The question asks to show using the residue theorem that \int cos(x)/(x2+1)2 dx = \pi/e (the terminals of the integral are -\infty to \infty but i didnt know the code to write that) I found the singularities at -i and +i so i think we can then say \intcos (z) / (z+i)2(z-i)2 dz...

If f is bounded on [a,b], can one define a Riemann-Stieltjes integral \int_a^b f(x) d\alpha(x) when the function \alpha(x) is not monotonically increasing on [a,b]? Rudin only seems to define R-S integrals with respect to monotonically increasing functions, but there are sources I've found...

Homework Statement \int|e-k|x-a||2dx This integral is from -inf to +inf Homework Equations A table of integrals would seem to be needed. The Attempt at a Solution \int|e-k|x-a||2dx =\int|e-2k|x-a||dx from here it would seem to simplify down to a variation on the theme of \int|e-ax|dx...

Homework Statement \begin{equation} \int_{-1}^{1} e^{u+1} \end{equation} Homework Equations The Attempt at a Solution I really seem to struggle with any problems with e in them. I think I may have missed some of the basic rules or something, but I can't seem to find what I missed...

Homework Statement I=\int_{-\infty}^{\infty} { dx \over {5x^2+6x+5}}Homework Equations The residue theorem.The Attempt at a Solution I can't use the residue theorem since the denominator has real zeros. How should I solve this?

Hi, I saw a proof/argument done today that I think was wrong: It is finding the limit as a->oo of the integral from 0 to b<oo: Int_(0..b) Sqr[x(1 +cos(ax))]dx , where Sqr is the square root Now, the argument given was that one could find a bound for the oscillation of...