Integrals Definition and 1000 Threads

Homework Statement A swimming pool has dimensions 25 m x 10 m x 3m (length x width x height.) When it is filled with water, what is the force on the bottom (Fb) of the pool? On the long side (Fl)? On the short sides (Fw)? (note that integrals are required.) If you are concerned with whether or...

I'm doing calclus 3 right now and I'm trying to put together the results of integrals. Can you correct me if I'm wrong and the one's I missed ( particularly 4 / 5 / 6). I also that the integrals can mean different things based on context. But in terms of areas and volumes atleast? 1) ∫ dx...

How do you integrate f(z) = e^(1/z) in the multiply connected domain {Rez>0}∖{2} It seems like integrals of this function are path independent in this domain since integrals of e^(1/z) exist everywhere in teh domain {Rez>0}∖{2}. Is that correct?

Homework Statement Find the volume of the solid. Under the paraboloid z = x^2 + y^2 and above the region bounded by y = x^2 and x = y^2 Well, those curves only intersects in the xy-plane at (0,0) and (1,1), and in the first Quadrant, and in that first Quadrant y = sqrt(x), and over that...

Homework Statement Evaluate Integrate (2-3x/(Sqrt.(1 - x^2))) dx Homework Equations 1/Sqrt.(1-x^2) = arctan The Attempt at a Solution I am so lost, but this is what I've tried, but didn't work... I separated the integral into two so Integral of (2/(Sqrt.(1-x^20))) dx - integral of...

In peskin p. 192, they says that the denominator (that is equation 6.43) is symmetric under x<--> y. Thay all so say that you can see it in equation 6.44. But one of the terms in the denominetor is y*q which dose not have that symmetry! Looking at (6.43) and removing the summetric parts leave...

Homework Statement a rigid body with a mass of 2 kg moves along a line due to a force that produces a position function x(t)= 4t^2, where x is measured in meters and t is measured in seconds. Find the work done during the first 5 seconds in two ways. Homework Equations x(t)= 4t^2 Work is ->...

Homework Statement My question is this: When finding center of mass, can you do so using spherical/cylindrical coordinates, or must you put it in cartesian coordinates? If you can use spherical/cylindrical coordinates, how do you set up the triple integrals ? Thank you. Homework...

Homework Statement Prove the Mean Value Theorem for integrals by applying the Mean Value Theorem for derivatives to the function F(x) = \int_a^x \, f(t) \, dt Homework Equations [/B] Mean Value Theorem for integrals: If f is continuous on [a, b], then there exists a number c in [a, b]...

I am just starting to learn double and triple integrals. Say: I = ∫ ∫ dx dy This should give the area within two curves right? What will the following integral give? Will it give volume? I am finding questions where it gives (for instance) mass in a given shape I = ∫ ∫ f(x,y) dx dy

Homework Statement (3 i) Using \nabla . \mathbf{F} = \frac{\partial \mathbf{F_{\rho}}}{\partial \rho} + \frac{\mathbf{F_{\rho}}}{\rho} + \frac{1}{\rho} \frac{\partial \mathbf{F_{\phi}}}{\partial \phi} + \frac{\partial \mathbf{F_{z}}}{\partial z} calculate the divergence of the vector field...

Homework Statement I know that a single integral can be used to find the area under a y = f(x) curve, but above the x axis. Correct me if this example of a double integral is invalid: If I hold a piece of paper in mid air and it droops, the double integral will give me the volume of the object...

Homework Statement Can a single integral be used to solve a multi-variable equation, and can a triple integral be used to find the area under an y = f(x) curve? What I'm getting at is whether or not single, double and triple integrals must be integrated with respect to their corresponding...

Homework Statement (hebrew) : f(x) a continuous function. proof the following Homework Equations I guess rules of limits and integrals The Attempt at a Solution I've tried several approaches: taking ln() of both sides and using L'Hospitale Rule. Thought about using integral reduction...

Hey! :o In my notes there is the following example about the energy method. $$u_{tt}(x, t)-u_{xxtt}(x, t)-u_{xx}(x, t)=0, 0<x<1, t>0 \\ u(x, 0)=0 \\ u_t(x, 0)=0 \\ u_x(0, t)=0 \\ u_x(1, t)=0$$ $$\int_0^1(u_tu_{tt}-u_tu_{xxtt}-u_tu_{xx})dx=0 \tag 1$$ $$\int_0^1...

I'm in the middle of the Great Courses Multivariable Calculus course. A double integral example involves a quarter circle, in the first quadrant, of radius 2. In Cartesian coordinates, the integrand is y dx dy and the outer integral goes from 0 to 2 and the inner from 0 to sqrt(4-y^2). In...

Hi! I have a dataset that I fit to a 5th order and 4th order polynomial -- I was just trying to get the function that best fit the data. However, I realized that when I evaluate the integral for these 2 different functions (between 200 and 400), the answers are vastly different. I assumed...

Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path integrals through a space with holes in it - with regions of space excluded from the integration...

Lately, I have been trying really hard to understand the Newton Leibnitz Formula for evaluating Definite Integrals. It states that- If f(x) is continuous in [a,b] then \int_a^b f(x) dx = F(b) - F(a). But one thing that just doesn't make sense to me is that why should f(x) be continuous in...

Homework Statement Find ∫ f(x) dx between [4,8] if, ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4 Homework Equations [/B] ∫ f(x) dx between [4,8] , ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4 The Attempt at a Solution We are given ∫ f(2x) dx between...

Homework Statement A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals. Homework Equations How to derive it The Attempt at a Solution I only figured out that all of this is related to the conservation of energy, but i don't know even the...

Homework Statement 2. Homework Equations [/B] Uploaded as a picture as it's pretty hard to type out The Attempt at a Solution So to normalise a wavefunction it has to equal 1 when squared. A is the normalisation factor so we have: A.x2e-x/2a0.x2e-x/2a0 = 1 ∫ψ*ψdx = A2∫x4e-axdx = 1 Then I've...

Consider an integral of the type ## \int_0^{a} \int_0^{\pi} g(\rho,\varphi,\theta) \rho d\varphi d\rho ##. As you can see, the integral is w.r.t. cylindrical coordinates on a plane but the integrand is also a function of ##\theta## which is a spherical coordinate. So for evaluating it, there are...

Homework Statement Its more of a general issue of understanding than a specific problem I have to evaluate a few surface integrals and I am not sure about the geometric significance of what I am evaluating or even of what to evaluate. Examples. If n is the unit normal to the surface S...

Hello everyone! I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the upper half side, but I don't know how to solve them when the singularities are on...

my question is this: you know than in feynman path integra, you integrate eiS/hbar along all the fields. you also know that S is real and that it is the integral of local functions (fields and derivatives of fields). you also know that path integral generates an unitary and local...

Hi! Here is my task: Calculate surface area of sphere $$x^{2}+y^{2}+z^{2}=16$$ between $$z=2$$ and $$z=-2\sqrt{3}$$. Here are 3D graphs of our surfaces: Surface area of interest is P3. It would be P-(P1+P2), where P is surface area of whole sphere. Is it correct? Here is how I calculated...

Homework Statement Here are the problems http://imgur.com/a/kbtPS The problems I need help with are 1 and 4(a) Homework Equations The second fundamental theorem of calculus The Attempt at a Solution For problem 1, I calculated the areas under the curve (using remmien summs) and tried to find...

Homework Statement Solve the integral equation for y(x): y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt (integral from 0 to x) See attached image for the equation in a nicer format. Homework Equations Fundamental Theorem of Calculus The Attempt at a Solution dy/dx = y(x)^2 / (1 + x^2) ∫ dy/y^2 = ∫...

When I learned Integrals in Calc III, the formula looked like this ∫∫ F(r(s,t))⋅(rs x rt)*dA but in physics for Gauss's law it is ∫∫E⋅nhat dA How are these the same basic formula? I know that nhat is a unit vector, so it is n/|n|, but in the actual equation, it is a dot between the cross...

Homework Statement Find the area enclosed by the equations: y=1/x and y=1/x^2 and x=2 Homework Equations N/A The Attempt at a Solution So I solved this analytically after looking at a graph of the two functions. Using integrals I got the following: ln(2)-1/2 Which is the correct answer. I...

Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...

Homework Statement Homework Equations Down The Attempt at a Solution As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the beginning n=1 the residue...

I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function? If an integral is the area under a curve, why do we even have to find the anti derivative...

I had a couple of questions. 1. Why does the integral ∫exf(t) dt transform to ex∫f(t) dt? Shouldn't ex be a part of the integrand too? 2. Why is the difference dy - dy1 = d(y - y1)?

Thank you :)

I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is- The indefinite integral of a function x with respect to f(x) is another function g(x) whose derivative is f(x). i.e. g'(x) = f(x) ⇒ Indefinite...

I actually came across this question on social media. What is: $$\int sin (x) \, dx - \int sin (x) \, dx$$ And I think the answer depends on how we interpret: $$\int sin (x) \, dx$$ If we think of it as a single antiderivative, the answer would be zero. If we think of it as being...

Homework Statement I am working through some maths to deepen my understanding of a topic we have learned about. However I am not sure what the author has done and I have copied below the chunk I am stuck on. I would be extremely grateful if someone could just briefly explain what is going on...

I saw this method of calculating: $$I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx$$ http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx Can you take a look at M.N.C.E.'s method? I don't understand a few things. Somehow he makes the relation...

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ $R$ is the big radius, $\delta$ is the small radius. Actually, let's consider $u$ the small radius. Let $\delta = u$ Ultimately the goal is to let $u \to 0$ We can parametrize, $$z =...

Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...

Note: I'm posting this in the Quantum Physics forum since it doesn't really apply to HEP or particle physics (just scalar QFT). Hopefully this is the right forum. In Peskin and Schroeder, one reaches the following equation for the spacetime Klein-Gordon field: $$\phi(x,t)=\int...

Hello, I was just wondering, we have what could be called the indefinite derivative in the form of d/dx x^2=2x & evaluating at a particular x to get the definite derivative at that x. But with derivation, we can algebraically manipulate the limit definition of a derivative to actually evaluate...

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...