Integrals Definition and 1000 Threads

Homework Statement Use 1) \frac{1}{2\pi}\int\limits_{-\pi}^{\pi} \frac{r_0^2 - r^2}{r_0^2 - 2rr_0cos(\theta-t) + r^2} dt = 1 to compute the integral: 2) \int\limits_{-\pi}^{\pi} \left[1 - acos(x) \right]^{-1} dx for 0<a<1 [/itex]. The Attempt at a Solution I looked on Wolfram...

Homework Statement Evaluate the integral \int\int_S \sqrt{1+x^{2}+y^{2}}dS where S:{ r(u,v) = (ucos(v),usin(v),v) | 0\leq u\leq 4,0\leq v\leq 4\pi } 2. The attempt at a solution Here is my attempt, I am fairly sure I am right, but it is an online assignment and it keeps telling me I am...

Homework Statement I understand everything except why the derivative of 2 + tan (x/2) is .5 + sec^2(x/2) I don't understand the .5 part. I understand the sec part. I would think the derivative of 2 would be C, or just disappear.

Here's an eclectic bunch.1) $ \displaystyle \int_{0}^{\infty} \frac{x^{2}}{(1+x^{5})(1+x^{6})} \ dx $2) $ \displaystyle \int_{0}^{\infty} \frac{1}{(1+x^{\varphi})^{\varphi}} \ dx $ where $\varphi$ is the golden ratio3) $ \displaystyle \int_{0}^{\infty} \sin \left(x^{2} + \frac{1}{x^{2}} \right)...

Homework Statement OK, this is something that stumped me. Homework Equations \int_{}^{}{\frac{dZ}{ A^{2}-Z^{2}}} = - \int_{}^{}{\frac{dZ}{ Z^{2}-A^{2} }} Right? \int_{}^{}{\frac{dZ}{ A^{2}-Z^{2} }}=\; \frac{1}{2A}\ln \left\{ \frac{A+Z}{A-Z} \right\}+\mbox{C} \int_{}^{}{\frac{dZ}{...

Homework Statement The Attempt at a Solution My book is not explaining very well the steps at solving these problems. There is a step that I'm missing step 1. find 1/(b-a) easy step 2. find the antiderivative of 4 - x, easy, x^2/2 step 3. plug in what the result of the...

Prove that: \[ \int_{0}^{\pi/2} \cos(nx) \cos^n(x) dx =\frac{\pi}{2^{n+1}}\] \[ \int_{0}^{\pi} \frac{1-\cos(nx)}{1-\cos(x)} dx =n\pi \] where \( n \in \mathbb{N} \). You can use induction, contour integration or any other method you like.

Fun! Fun! Fun! Here are more entertaining problems: 1.\( \displaystyle \int_{2}^{4} \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(3+x)}+\sqrt{\ln(9-x)}}dx\) 2.\( \displaystyle \int_{\sqrt{\ln(2)}}^{\sqrt{\ln(3)}}\frac{x \sin^2(x)}{\sin(x^2)+\sin(\ln(6)-x^2)}dx\) 3.\( \displaystyle...

By parameterizing the curve (not by Cauchy's theorem) and using the series of sin z, nd the value of ∫z^k sin(z)dz around a closed Contour C where C is the unit circle z=e^(iθ), for 0≤θ<2π What do they mean by using series of sin z ? I mean if I expand it .. I get e^(iθ)- e^(3iθ)/3! --- and...

Homework Statement Let f be a continuously differentiable function on the interval [0,2\pi], where f(0) = f(2\pi) and f'(0) = f'(2\pi). For n = 1,2,3,\dotsc, define a_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) dx. Prove that the series \sum_{n=1}^\infty |a_n|^2 converges...

Folks, 1) If we have \int F \cdot dr that is independent of the path, does that mean that the integral will always be 0? 2) For 2 dimensional problems when we evaluate line integrals directly and use Greens Theorem for every piece wise smooth closed curves C, arent we always calculating...

Homework Statement Evaluate this integral directly Homework Equations \int cos x sin y dx +sin x cos y dy on vertices (0,0), (3,3) and (0,3) for a triangle The Attempt at a Solution Does this have to evaluated parametrically using r(t)=(1-t)r_0+tr_1 for 0 \le t\le 1 or can I just...

how would one evaluate this without using trig substitution? Is it possible to make one integral out of this? {int[(y^2 + a1^2)^-1]dy +c1}/{int[(x^2 + a2^2)^-1]dx +c2} +c3 the numbers behind the 'a's and 'c's are supposed to be subscripts. Also, how would one deal with this...

Hi, I'm studying calculus 3 and am currently learning about conservative vector fields. ============================= Fundamental Theorem for Line Integrals ============================= Let F be a a continuous vector field on an open connected region R in ℝ^{2} (or D in ℝ^{3}). There exists...

Folks, When we are evaluating integrals like the following, what are we evaluating in terms of units etc. For example if I integrate Fdx I get an area which represents the energy where F is the force and d is the displacement so the units are Nm etc. 1) Integrals over intervals ...

Alright, I have a kind of dumb question: Why do I distinguish between dq and dqi when considering the propagation from qi to q to qf? For example, if we want the wave function at some qf and tf given qi and ti, we may write: ψ(qf,tf)=∫K(qftf;qiti)ψ(qi,ti)dqi Why do we distinguish between dqi...

In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of...

hi folks, one often reads \int_A f(x) dx = \int_A g(x) dx for arbirary A, thus f(x) = g(x), since the equaltiy of the Integrals holds for any domain A. I don't see, why the argument "...for any domain A..." really justifies this conclusion. Can someone explain this to me, please?

Alright, so I was wondering if anyone could help me figure out from one step to the next... So we have defined |qt>=exp(iHt/\hbar)|q> and we divide some interval up into pieces of duration τ Then we consider <q_{j+1}t_{j+1}|q_{j}t_{j}> =<q_{j+1}|e-iHτ/\hbar|q_{j}>...

There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D...

Homework Statement Interpret the integrals (from 0 to 4)∫ (3x/4) dx + (from 4 to 5)∫ (sqrt(25-x^2)) dx as areas and use the result to express the sum above as one definite integral. Evaluate the new integral. Homework Equations The Attempt at a Solution I see that I could...

Find the surface area of the triangle with vertices (0,0) (L, L) (L,-L) I know I have to take the double integrals of f(x,y) but I have no idea what f(x,y) is supposed to be!

Homework Statement Prove \sqrt{\frac{2}{\pi}}\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=t^{-\frac{1}{2}} and use that to solve \int^{\infty}_0\cos y^2dy Is this good way to try to prove? Homework Equations The Attempt at a Solution Homework Statement...

derivative of integral over e^t to t^5 (sqrt(8+x^4)) dx I know I need to use the chain rule and I can take the derivative of the integral without respect to e^t and t^5. If you know the answer, can you answer and tell me how to do it?! Calculus final on Monday...

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...

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...

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...

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...