Integrals Definition and 1000 Threads

I have always seen the integral as the area under a curve. So for instance, if you integrated over the upper arc of a circle you would get ½\piR2. But then, I learned to do integrals in spherical coordinates and something confuses me. If you do the integral from 0 to 2\pi, you don't get the...

Homework Statement Find the mass of a solid of constant density that is bounded by the parabolic cylinder x=y2 and the planes x=z, z=0, and x=1. The Attempt at a Solution https://dl.dropbox.com/u/64325990/Photobook/Photo%202012-06-07%202%2033%2024%20PM.jpg I first drew some diagrams to...

Homework Statement A lamina occupies the region inside the circle x2+y2=2y but outside the circle x2+y2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin. Here is the solution...

Homework Statement show that \int^{∞}_{0}\frac{sin^{2}x}{x^{2}}dx= \frac{\pi}{2} Homework Equations consider \oint_{C}\frac{1-e^{i2z}}{z^{2}}dz where C is a semi circle of radius R, about 0,0 with an indent (another semi circle) excluding 0,0. The Attempt at a Solution...

Hello again. I am hoping that someone can assist in checking my work regarding a trigonometric integral. The problem and my attempt to solve is as follows. \int\sin^{\frac{-3}{2}}(x)*cos^3(x) dx \int\sin^{\frac{-3}{2}}(x)*cos^2(x)*cos(x) dx Using a Pythagorean identity...

Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/Photobook/question.PNG Here is the answer: http://dl.dropbox.com/u/64325990/Photobook/solution.PNG So what the answer says is that you can find the volume under the surface minus the volume of the rectangle with height...

I heard that the formula below can be used to evaluate some kinds of integrals but I can't find what kinds and how to do it.Could someone name those kinds and also the procedure? \frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) b'(x) - f(x,a(x)) a'(x) + \int_{a(x)}^{b(x)}...

I'm preparing for a vector calculus course in the fall and I've been self studying some topics. I've taken multivariable calculus and I'm familiar with using double integrals, how to solve them and how to use them to find volume. From what I've read so far, I'm familiar with how to SOLVE a...

Homework Statement I'm just interested in knowing where the 4 comes from in front of the integral.

Homework Statement Do you see how y gets converted to csc? I don't get that. I would y would be converted to sin in polar coordinates.

Homework Statement do you see how the integral of r is .5? I don't get how that follows?

Homework Statement The Attempt at a Solution I understand the steps, although it took quite a while, but what I don't understand is that a triangle with base 2 and height 2, it's area is 2. With two triangles of that size the area should be 4. The books says the area is 8.

Hi everyone, Say I have a 2D one loop integral of the form $$ I^s_n(\Delta)=\int d^2 l \frac{(l^2)^s}{(l^2-\Delta)^n} $$ Using that ##1 = \frac{1}{2} \partial_\mu l^\mu##, I can relate say $$I^1_2(\Delta)=I^0_1(\Delta)$$ + total derivative term. In dimensional regularization one usually...

Let f : R to R be a continuous function, and suppose that definite integral from m to n |∫(m to n)f(x)dx|≤(n-m)^2 for every closed bounded interval [m, n] in R. Then is it the case that f(x) = 0 for all x in R? I tried using fundamental theorem of calculus but got stuck, since I only got that...

Homework Statement If the power of the secand is even and positive.. \int sec^{2k} x tan^{n} x dx = \int (sec^2 x)^{k-1} tan ^n x sec^2 x dx The Attempt at a Solution The way I see it, sec^{2k} x = sec^2 x dx * sec^k x dx the next step seems to be to break down sec^k, but on closer...

*SOLVED* Indefinite integrals. Arriving at different results. Mistake found. Thanks!, everything looks correct now.

Homework Statement Compute the surface integral for F = [3x^2, y^22, 0] and S being a portion of the plane r(u,v)=[u,v,2u+3v], 0≤u≤2, −1≤v≤1.The Attempt at a Solution I managed to get the correct answer, because with some luck I defined the normal in the correct direction. I am just confused...

The integral for calculating the flux of a vector field through a surface S with parametrization r(u,v) can be written as: \int\int_{D}F\bullet(r_{u}\times r_{v})dA But what's to stop one from multiplying the normal vector r_{u}\times r_{v} by a scalar, which would result in a different...

Help choose the limits of the following volume integrals: 1) V is the region bounded by the planes x=0,y=0,z=2 and the surface z=x^2 + y^2 lying the positive quadrant. I need the limits in terms of x first, then y then z AND z first, then y and then x. And also polar coordinates, x=rcost...

Homework Statement Find the area in the positive quadrant of the x-y plane bounded by the curves {x}^{2}+2\,{y}^{2}=1, {x}^{2}+2\,{y}^{2}=4, y=2\,x, y=5\,x The Attempt at a Solution This is a graph of the region: http://img21.imageshack.us/img21/2947/59763898.jpg One thing I was...

Calculate the following contour integrals using suitable parameterisations Homework Statement 1)##\oint \frac{1}{z-z_0} dz## where C is the circle ##z_0## and radius r>0 oriented CCW and ##k\ge0## 2) ##\int_c |z|^2 dz## where C is the straight line from 1+i to -1 3. Relevant equations...

Homework Statement Calculate the following contour integrals \int_{c1} (x^3-3xy^2 ) + i (3yx^2 - y^3) where c1 is th line from 0 to 1+i Homework Equations The Attempt at a Solution a earlier part of the question asked if it was analytic. using Cauchy-Reimann equations i have...

Homework Statement a) \int\int_{B}\frac{\sqrt[3]{y-x}}{1+y+x} dxdy, where B is the triangle with vertices (0, 0), (1, 0), (0, 1). b)\int\int_{B}x dxdy where B is the set, in the xy plane, limited by the cardioid ρ=1-cos(θ) The Attempt at a Solution a) Let ψ: \left\{u = y-x, v =...

Weinberg in his 1st book on QFT writes in the paragraph containing 2.5.12 that we may choose the states with standard momentum to be orthonormal. Isn't that just true because the states with any momentum are chosen to be orthonormal by the usual orthonormalization process of quantum mechanics...

Homework Statement Evaluate the follow by first changing the order of integration \int_{x=-1}^{1}\int_{y=x^2}^{2-x^2}dydxThe Attempt at a Solution This is the region we're concerned with: http://www.wolframalpha.com/input/?i=plot%28y%3Dx^2%2C+y+%3D+2+-+x^2%2C+x%3D+1%2C+x%3D+-1%29 The new...

Hi, I am able to manipulate and use double integrals, but I am having a bit of mental block when trying to visual how they actually work. First, would you agree that a double integral is simply summing a function over a region by taking lots of tiny squares (or rectangles?) of sides dx...

Homework Statement Let C be the boundary of the region bounded by the curves y=x^{2} and y=x. Assuming C is oriented counter clockwise, Use green's theorem to evaluate the following line integrals (a) \oint(6xy-y^2)dx and (b) \oint(6xy-y^2)dyHomework Equations The Attempt at a Solution...

Homework Statement Evaluate the integral Homework Equations I can substitute and thus end up with The Attempt at a Solution I then expand the denominator out and end up with 1/ However I then assume I need to factorise the top line of that fraction as this will be the...

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 Let J0(x)=2/\pi\int0\pi/2cos(xcos[y])dy. Show that \int0∞J0(x)e-axdx=\frac{1}{sqrt(1 +a^2)}. Homework Equations Tonelli and Fubini's theorems The Attempt at a Solution Basically I'm finding this problem really hard because I've had to teach myself iterated...

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...