Integrals Definition and 1000 Threads

Homework Statement Suppose f : ℝ→ℝ and g : ℝ →ℝ are continuous. Suppose that f is odd and g is even. Define h(x,y) : f(x)*g(y). Let D be a disk centered at the origin in the plane. What is ∫∫h(x,y)dA? D The Attempt at a Solution I know there's probably a trick to it. Is it 0...

Homework Statement I'm a bit confused as to how to determine which component must be positive or negative if the question gives you a surface and says the normal vector is pointing outward or inward. Some surfaces have it so that the z component is positive if n is pointing outward and...

\oint_S \vec{A}\cdot d\vec{S}=\int_V div\vec{A}dv Suppose region where \vec{A}(\vec{r}) is diferentiable everywhere except in region which is given in the picture. Around this region is surface S'. In this case Gauss theorem leads us to \int_S \vec{A}\cdot d\vec{S}+\int_S \vec{A}\cdot...

Homework Statement I'm a bit uncertain as to how to do these types of integrals. Let γ be any contour from 1 - i to 1 + i. Evaluate the following: ∫ 4z^3 dzThe Attempt at a Solution I did this in three different methods, two of them gave the correct answer, although this could just be a...

Evaluate ∫∫xexp(xy)dA, and R (over which the integrand is to be integrated) is {(x,y)|0≤x,y≤1}. Could someone explain how this is to be done.

Does anyone know a simple proof for holder's inequality? I would be more interested in seeing the case of |∫fg|≤ sqrt(∫f^2)*sqrt(∫g^2)

Homework Statement Find the antiderivative of (x*arctan(x))/(1+x^2)^2) The Attempt at a Solution I've had a few attempt at this (I've been working on it an embarrassingly long time) but i felt most on track doing it by parts. Here's how i went u = arctan(x) du = 1/(1+x^2)*dx...

State the Fundamental Theorem: Let F be a vector field. If there exists a function f such that F = grad f, then \int_{C} F \cdot dr = f(Q) - f(P) where P and Q are endpoints of curve C. _________________________________ I didn't receive any credit for this answer. Admittedly...

I'm having a lot of trouble with the subject. Here's one example I'd like explained. F(t_1, t_2) = \int \limits_0^1 x^{t_1}\ln^{t_2}\frac{1}{x} dx The book asks to find for what \vec{t} F converges. The answer is \vec{t}\in(-1; \infty)^2, but I don't see how to get that. In general, what...

Vector field F(bar)= <6x+2y,2x+5y> fx(x,y)= 6x+2y fy(x,y)= 2x+5y f(x,y)= 3x^2+2xy+g(y) fy(x,y)=2x+g'(y) 2x+g'(y)= 2x+5y g'(y)= 5y g(y)= 5/2*y^2 f(x,y)=3x^2+2xy+(5/2)y^2 Then find the \int F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1 I'm stuck on finding the last part for the F(bar)...

Prove the following: If f is Riemann integrable on an interval [a,b], show that ∀ε>0, there are a pair of step functions L(x)≤f(x)≤U(x) s.t. ∫_a^b▒(U(x)-L(x))dx<ε My proof: Since f is Riemann integrable on [a,b] then, by Theorem 8.16, ∀ε>0, there is at least one partition π of the interval...

Homework Statement Suppose \int_{-\infty}^{\infty}t|f(t)|dt < K Using Cauchy-Schwartz Inequality, show that \int_{a}^{b} \leq K^{2}(log(b)-log(a)) Homework Equations Cauchy Schwartz: |(a,b)| \leq ||a|| \cdot ||b|| The Attempt at a Solution Taking CS on L^{2} gives us...

So kind of like this thread, I'm looking to convert a discrete sum to an integral. My idea thus far has been to arrive at a function via spline interpolation. I'm doing a few different types of sums, but the first ones look like \displaystyle a=\sum_{i=1}^{100}{data[1]*data[4]} where data...

Homework Statement The depth of water in a swimming pool fits the equation f(x,y) = 2sin (x/20 - 7) - 3 cos ( x-3 /5)+8 when 0<=x<=20 and the sides of the pool fir the equations y(x) = 10-(x-10)^2/10 and y(6)= (x-10)^2/20 -5 Find the volume of the water in the pool using a double integral...

Homework Statement Use a triple integral to find the volume of the region. Below x+2y+2z=4, above z=2x, in the first octant. Homework Equations V=∫∫∫dV=∫∫∫dxdydz The Attempt at a Solution I have no clue where to begin as to finding those darn limits to integrate with. I'm sure...

Homework Statement a) S 13(4^x + 3^x)dx b) S (cosx + sec^2x)dx c) S (3-(1/x))dx d) S e^(7x)dx Homework Equations The S is supposed to be the integration sign The Attempt at a Solution Are these correct or at least close? a) = 13((4^x)/(ln(4) + (3^x)/(ln(3))) + C b) =...

Homework Statement Use Polar coordinates to evaluate were C denotes the unit circle about a fixed point Z0 in the complex plane The Attempt at a Solution I've only used polar integrals to convert an integral in sin and cos into one in therms of z, find the residues and then use the...

I've started self-teaching asymptotic methods, and I have some theoretic questions (and lots of doubts!). 1. Say I have the asymptotic expansion f(x) \asymp \alpha \sum_n a_n x^{-n} for x large, where \alpha is some prefactor. How can I estimate the value of n for the term of...

other than the physics (work) what are the applications of line integral? particularly does it have any use in finance or economics?

Homework Statement I am currently taking calc III and we have starting getting into double and triple integrals. I was wondering what you are actually doing when you take a double or triple integral? And what the difference is. I understand that you find area with a single integral and find...

Homework Statement http://img710.imageshack.us/img710/4764/doubleintegral.png Homework Equations The Attempt at a Solution Now, my understanding of the region is that x spans from the line x = y to x = 1, and that given that parameter, the applicable y's are 0 to 1. In other...

Homework Statement Find the mass m of the pyramid with base in the plane z = 9 and sides formed by the three planes y = 0 and y - x = 5 and 6x + y + z = 28, if the density of the solid is given by δ(x,y,z) = y. Homework Equations The Attempt at a Solution This problem is driving...

I'm curious about the validity of various techniques from good old calculus in one real variable when dealing with complex coefficients. I know enough complex analysis to know that the rules change when dealing with complex variables, but I'm curious about the case when the variables are still...

Homework Statement \int_{|z-2i|=2} = \frac{dz}{z^2-9} 2. The attempt at a solution I know that the contour described by |z-2i|=2 is a circle with a center of (0,2) (on the complex plane) with a radius of 2. The singularities of the integral fall outside of the contour (z+3 and...

Homework Statement Im righting this down for my roommates since he's having tons of trouble trying to figure this out and I can't answer it. also sorry for having to hotlink it. http://i.imgur.com/afShz.jpg the equation is on the image since its very difficult to type it all out...

Interpretation of "dx" as the differential of x for Indefinite Integrals This question is concept-as-opposed-to-calculation based. I understand that when one sees the integral sign, followed by f(x)dx, that we can think of this as the indefinite integral, or antiderivative of f(x), with...

Homework Statement http://img404.imageshack.us/img404/3952/contf.png The Attempt at a Solution Is there a set of rules or postulate that refer to which contour to use for specific integrals? I tried to use the residue theorem for the first integral but I didn't get the right answer

integrals (lots of them!) 1. $\displaystyle \int_{0}^{1}\frac{\sin(\ln x)}{\ln x}dx$ 2. $\displaystyle \int_{0}^{1}\frac{\arctan(x)}{1+x}dx$ 3. $\displaystyle \int_{0}^{\infty}\frac{\ln(1+x^2)}{1+x^2}dx$ 4. $\displaystyle \int_{0}^{\infty}e^{- \left( x^2+\dfrac{1}{x^2}\right)}dx$ 5...

I'm working on an online EECS course, and to be frank some of it is going straight over my head - but at the same time parts of it are far below my current knowledge, so I want to work and stick with it. The speaker is working through proving current and voltage - to arrive at Kirchoff's...

Homework Statement ∫e2xarctan(ex)dx Homework Equations From the table of integrals: #92 ∫utan-1udu = (u2+1)/2)tan-1-u/2 + c or #95 ∫untan-1udu = 1/(n+1)[un+1tan-1-∫ (un+1du)/(1+u2) , n≠-1 The Attempt at a Solution The answer is 1/2(e2x+1)arctan(ex) - (1/2)ex + C I don't...

Homework Statement Show that T(u,v) = (u2 - v2, 2uv) maps to the triangle = {(u,v): 0 ≤ v ≤ u ≤ 4} to the domain D bounded by x=0, y=0, and y2 = 1024 - 64x. Use T to evaluate ∬D sqrt(x2+y2) dxdy Homework Equations The Attempt at a Solution x=u2-v2 y=2uv Jacobian= 4u2+4v2 dudv I guess the...

The ratio of integrals ∫〖a(x) b(x)dx〗/ ∫c(x) b(x)dx can be maximized by choosing b(x) equal to the delta function at the point where a(x)/c(x) is a maximum. Can anyone provide the solution for choosing b(x) when b(x) cannot equal the delta function, b(x) is greater than zero with a...

Homework Statement do my professor did this in class, and it doesn't make sense to me ∫cos^5(x) sin^4(x) dx ∫cos^4(x) sin^4(x) cos(x) dx ∫(1-sin^2(x))^2 sin^4(x) cos(x) dx ∫(sin^4(x) -2sin^6(x) +sin^8(x))cos(x) dx so the above I get but when my professor integrated I became lost, this...

1. The problem statement, all variables and given/known data Calculate the given iterated integrals ∫02 dy ∫0yy2 * exy dxMy attempt: ∫20dy[exy*y]y0 = ∫20 ey*y*y - ex*0*0 = ∫20ey2*y dx = [ey^2]*y]20 = 2e4 Is this correct?

Homework Statement Evaluate the intergrals: a) integral of 3^(-4*z^2) dz from 0 to infinity b) integral of dx/(sqrt(-ln(x))) from 0 to 1 c) integral of x^m * e^(-a*x^n) dx from 0 to infinity Homework Equations gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity The...

In "Probability Theory: The Logic of Science", the author E. T. Jaynes relates that Prof. Hansen at Stanford evaluated integrals by treating constants like pi in an integrand as a variable. Sounds fantastic! Does anyone know how this is done?

Homework Statement Find the volume of the solid enclosed between the cylinder x2+y2=9 and planes z=1 and x+z=5Homework Equations V=∫∫∫dz dy dzThe Attempt at a Solution The problem I have here is setting the integration limits. I first tried using: z from 1 to 5-x y from √(9-x2) to -3 x from -3...

Have you ever see any books discussing these problems? I don't know the name of these topic.

Improper Integrals -- Infinite Intervals Homework Statement Evaluate the integral. (from e to infinity) ∫(25/x(lnx)^3)dx Homework Equations The Attempt at a Solution I know that for evaluating improper integrals, you can take the limit as t approaches infinity of the given...

Homework Statement Suppose that p and q are points in U, where U is an open, path-connected, simply connected subset of Rn and c1 and c2 are smooth curves in Rn with c1(0)=c2(0)=p, c1(1)=c2(1)=q. Let w be a 1-form on U. Prove that the line integral of w over c1 equals the line integral of w...

Someone told me Isaac Newton developed some infinitesimal triangle series to find the area of a random blob, but I think there might be some way to do it this way by drawing many lines from a central point to the edge, although that would make more of a pie slice, but is there some way to...

Greetings chaps, This will probably be old hat to most of you, but I'm beginning to start Quantum mech. so that I can develop a deeper understanding of its application in Chemistry ( I'm a Chemistry undergrad -gauge my level from that if you will!) i.) First of all, would I be right in...

Homework Statement Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$ converges to M The Attempt at a Solution Where do I start? I'm thinking of having g_n(x)=...

Please help me to evaluate the following integrals: 1) $\int\frac{x^{4}+1}{x^{2}+1}dx$ I recognise the form of $x^{2}+1$ in the denominator corresponds to an inverse tangent derivative. But how would I deal with the numerator in this respect? 2) $\int\frac{1}{x^{2}+x-6}dx$ I believe this...

I see equations of the form, y=\int_{-\infty }^{t}{F\left( x \right)}dx a lot in my texts. What exactly does it mean? From the looks of it, it just means there is effectively no lower bounds. I looked up improper integrals, but I can't say I really understand what is going on. So when...

Ok, so for Q6, I first said that z = 3 - 3x - 1.5y Using (∂z/∂x)^2 = 9, (∂z/∂y)^2 = 9/4 I then did a double integral of (x + y + (3 - 3x - 1.5y)) * sqrt(9 + 9/4 + 1) dA Letting y and x be bounded below by 0 as stated, and x bounded above by 1 - 0.5y and y bounded above by 2, I went...

Been a long time I had my integral class so I forgot almost everything I knew... I need to integrate to see if the serie converge (limn→∞ an = 0). Thus, there is a theorem of the integral, if you evaluate the limit of the integral of a serie when it tends to the infinite minus when x=1 you can...

Hi, In my fluids work I have come to integrals of the type: \int_{0}^{\infty}\frac{e^{ikx}}{ak^{2}+bk+c}dk I was thinking of evaluating this via residue calculus but I can't think of the right contour, any suggestions? Mat

$\displaystyle \int_{0}^{1} \ln \ x \ dx $ is not a proper Riemann integral since $\ln \ x $ is not bounded on $[0,1]$. Yet $ \displaystyle \int_{0}^{1} \ln \ x \ dx = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left(\frac{k}{n} \right)$. Is this because $\ln \ x$ is monotone on $(0,1]$?

Hey people, I need to calculate inner product of two Harmonic oscillator eigenstates with different mass. Does anybody know where I could find a formula for \int{ H_n(x) H_m(\alpha x) dx} where H_n, H_m are Hermite polynomials?