# Integals Definition and 62 Discussions

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1. ### I Schrodinger Equation from Ritz Variational Method

(This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method) 1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
2. ### Trouble with a Rocket Propulsion question (Variable Mass & Momentum)

I chose to set the upwards direction to be positive and dM/dt = R = 190 kg/s, so I can solve the problem in variable form and plug in. With the only external force being gravity, this gives M(t) * dv/dt = -M(t) * g + v_rel * R where M(t) is the remaining mass of the rocket. Rearranging this...
3. ### A Analytical solution for an integral in polar coordinates?

Hi, I am trying to find open-form solutions to the integrals attached below. Lambda and Eta are positive, known constants, smaller than 10 (if it helps). I would appreciate any help! Thank you!
4. ### Arc length of vector function - the integral seems impossible

The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...
5. ### The center of mass of a semicircular arc of non-negligible width

My attempt: 1) I am going to start this with a goal of setting up a reimann sum. First I divide the "arc"(?) of angle pi into n sub-arcs of equal angle Δθ 2) The total center of mass can be found if centers of mass of parts of the system are known. In each circular arc interval, I choose a...
6. ### Error in trapezoidal integration using a Programming language

Summary:: I want to iterate a mathematical model using a programming language. The equation of the mathematical model is simple. The following is a brief explanation. I want to iterate a mathematical model using a programming language. The equation of the mathematical model is simple. The...
7. ### Need help deducing the region for this double integration problem

Converting to a polar integral : Integrate ##$$f(x, y)=$$ $$\left[\ln \left(x^{2}+y^{2}\right)\right] / \sqrt{x^{2}+y^{2}}$$## over the region ##$$1 \leq x^{2}+y^{2} \leq e$$## So, \begin{array}{c} 1 \leq x^{2}+y^{2} \leq e \\ 1 \leq x^{2} \leq e \quad 1 \leq y^{2} \leq e \\ 1 \leq x \leq...
8. ### A Algebra of divergent integrals

Hello, guys! I would like to know your opinion and discuss this extension of real numbers: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 In essence, it extends real numbers with entities that correspond to divergent integrals and series. By adding the rules...
9. ### A Square of an integral containing a Green's Function

Let's say you have a tensor u with the following components: $$u_{ij}=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'$$ Where G is a Green function, and g is just a normal well behaved function. My question is what is the square of this component? is it...
10. ### I How to find the maximum arc length of this equation?

After seeing a discussion about graphs of the relationship ##x^x + y^y = r^r##, it got me interested in attempting to see what the graphical appearance of ##{^{\infty}x}+{^{\infty}y}={^{\infty}r}## would look like. The first step I did was use the relationship of...
11. ### I Integral involving up-arrow notation

I was playing around with a graphing program and sketching polar graphs involving tall power towers, when I noticed that ##sin(\theta) \uparrow \uparrow a## has an alternating appearance depending on whether ##a## is odd or even. I also noticed that the area enclosed by these alternating graphs...
12. ### I Integration: When to multiply by one or add zero?

I have seen several functions be integrated by multiplying by a form of one or by adding a form of zero. When is it advantageous do do one of these things? Are there any example problems (Calc I or II) in which I can try these techniques?

Let I = ##\int{\sqrt{\frac{cosx - cos^3x} {1-cos^3x}}}\,dx## I = ##\int{\sqrt{\frac{cosx(1 - cos^2x)} {1 - cos^3x}}}\,dx## I = ##\int{\sqrt{\frac {cosx} {1 - cos^3x }}}sinx\,dx## Substitute ##cosx = t## Therefore, ##sinx\,dx = -dt## So, I = ##\int{-\sqrt{\frac {t} {1 - t^3}}}\,dt## I'm stuck...
14. ### Using Integrals to Calculate the Rotational Energy of Earth

So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that...
15. ### B Understanding the basics of integration

I tried learning calculus using the book by Spivak.In this text, while introducing integrals the author explained a lot about partitioning the area under the curve and defined the integral.The way I understood this is, as we increase the number of divisions in the partition the lower sum and the...
16. ### I Invert a 3D Fourier transform when dealing with 4-vectors

I am having trouble following a step in a book. So we are given that $$\varphi (x) = \int \frac {d^3k}{(2\pi)^3 2\omega} [a(\textbf{k})e^{ikx} + a^*(\textbf{k})e^{-ikx}]$$ where the k in the measure is the spatial (vector) part of the four-momentum k=(##\omega##,##\textbf{k}##) and the k in the...
17. ### How to get the integral result?

Homework Statement I am studying Gerry's <Introductory Quantum Optics>, in which there is an integral (Eq. 4.37) $$\intop_{-infinity}^{+infinity}\frac{[sin(\triangle t/2)]^{2}}{\triangle^{2}}d\triangle=\frac{\pi}{2}t.$$ I don't know how to get the result of the right side. Homework Equations I...
18. ### Problem with plotting a function in MATLAB

Homework Statement Write code for solving the integral ##\int_{0}^{x}e^{-t^2}dx## using simpsons method and then plot the function from ##x = 0## to ##x = 5## with ##0.1## increment. Homework Equations 3. The Attempt at a Solution [/B] I was told that the best way to plot the function is to...
19. ### Mathematica Cannot do the integral of the Hyper-geometric function?

Dear friends: It's strange that Mathematica can do the integral of ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##, however, fails when it's changed to ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##. Are there any major differences between this two types? Is it possible to do the second kind of integral...
20. ### A Lebesgue measure and integral

Hello. I have problem with this integral : \lim_{n \to \infty } \int_{\mathbb{R}^+} \left( 1+ \frac{x}{n} \right) \sin ^n \left( x \right) d\mu_1 where ## \mu_1## is Lebesgue measure.
21. T

### Show the Fourier transformation of a Gaussian is a Gaussian.

Homework Statement Show, by completing the square in the exponent, that the Fourier transform of a Gaussian wavepacket ##a(t)## of width ##\tau## and centre (angular) frequency ##\omega_0##: ##a(t)=a_0e^{-i\omega_0t}e^{-(t/\tau)^2}## is a Gaussian of width ##2/\tau##, centred on ##\omega_0##...
22. ### Stuck on the integration bound

Homework Statement Trying to help a friend with a problem. We are supposed to solve the below using polar coordinates. The actual answer is supposed to be π/16. Solving the integral is not the issue, just converting it. 2. The attempt at a solution What I got sort of worked, but it is only...
23. D

### Find the limit using Riemann sum

Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework Equations The Attempt at...

Homework Statement I need to simplify the following integral $$f(r, \theta, z) =\frac{1}{j\lambda z} e^{jkr^2/2z} \int^{d/2}_0 \int^{2\pi}_0 \exp \left( -\frac{j2\pi r_0 r}{z\lambda} \cos \theta_0 \right) r_0 \ d\theta_0 dr_0 \tag{1}$$ Using the following integrals: $$\int^{2\pi}_0 \cos (z... 25. ### A Time differentiation of fluid line integrals I am looking at a proof from a book in fluid dynamics on time differentiation of fluid line integrals - Basically I am looking at the second term on the RHS in this equation$$ d/dt \int_L dr.A = \int_L dr. \partial A / \partial t + d/dt \int_L dr.A$$The author has a field vector A for a... 26. ### Calculus Books to learn integration techniques ? Are there books that are solely devoted to solving integrals and different methods in solving them ? I like solving integrals and I want to learn different techniques to solve integrals. 27. ### A Coulomb integrals of spherical Bessel functions Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf).$$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...

Homework Statement Homework Equations Final answer please it's a nuclear engineering course. The Attempt at a Solution I got 2pie as a final answer
29. ### Work problem -- lifting water out of tanks

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > I've encountered 2 problems in a row that involve lifting water out of tanks and finding the work needed. I am getting the incorrect answer. w = ⌠ab pgA(y)D(y)dy here is one of the problems: A...
30. ### I Work Problem

"It takes 100J of work to stretch a spring 0.5m from its equilibrium position. How much work is needed to stretch it an additional 0.75m." Attempt: w = ⌠abF(x)dx work = F x D 100J = F x 0.5m F = 200J 0.75 + 0.5 = 1.25 w = ⌠0.51.25 200dx w = 150 J The correct answer: w = 525 J what did I do...
31. ### First order separable Equation ODE

Homework Statement \frac{dy}{dx}\:+\:ycosx\:=\:5cosx I get two solutions for y however only one of them is correct according to my online homework (see attempt at solution) Homework Equations y(0) = 7 is initial condition The Attempt at a Solution \int \:\frac{1}{5-y}dy\:=\:\int...
32. ### Integral equation

The problem I want to find ##x## which solves ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt = 0 ## The attempt ##\int^x_1 \frac{\sin t}{t} \ dt = x -1 ## I see that the answer is ##x=1## but I want to be able to calculate it mechanically in case if I get similar problem with other elements. Any...
33. ### Step function integral

The problem I want to calculate ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx ## for the step function below. The attempt I started with rewriting the function as with the help of long-division ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx = \int^6_{-6} 1 \ dx - 2\int^6_{-6} \frac{1}{g(x)+2} \ dx## I...
34. ### Green's Function and integral

Homework Statement Find the green's function for y'' +2y' +2y = 0 with boundary conditions y(0)=y'(0)=0 and use it to solve y'' + 2y' +2y = e^(-2x) Homework Equations ##y = \int_a^b G(x,z)f(z)dz## The Attempt at a Solution I'm going to rush through the first bit. If you need a specific step...
35. ### Solving for Cn to get wave function

I need to solve Cn for a wave function, and have reached the following integral: Cn = -[√(1/a)](a/nπ)[cos(nπx/a)(ψ1(x)+ψ2(x))+∫cos(u)(dψ1(x)/dx)dx+∫cos(u)(dψ2(x)/dx)dx] This is a simplified version of the original equation, for elaboration Cn is the constant for linear combinations of a wave...
36. ### I Complex integral problem

I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...
37. ### Contour integral using residue theorem

Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...
38. ### I Can anyone evaluate this integral?

Hello All! I am trying to solve the simple pendulum without using a small angle approximation. But I end up with this integral: $$\int_{\frac{\pi}{4}}^{\theta}\frac{d\theta}{\sqrt{cos(\theta)-\frac{\sqrt{2}}{2}}}$$ Is this possible to evaluate? If so, could I get a hint about what methods to...
39. ### Volume of revolution around y-axis....stumped

Homework Statement The total area between a straight line and the parabola is revolved around the y-axis. What is the volume of revolution? According to the book, the answer is ; My answer comes out to be Homework Equations The Attempt at a Solution 1. Rewrite the second equation in...
40. ### B Using infinitesimals to find the volume of a sphere/surface

I've always thought of dxat the end of an integral as a "full stop" or something to tell me what variable I'm integrating with respect to. I looked up the derivation of the formula for volume of a sphere, and here, dx is taken as an infinitesimally small change which is multiplied by the area of...
41. ### B What would the "correct" way of doing this integral be?

v(x(t)), where v represents velocity and is a function of position which is a function of time. I have the equation: v dv/dx = 20x + 5 and want to solve for velocity. The way our professor solved it was by multiplying both sides by dx and integrating => ∫v dv = ∫20x+5 dx. I know doing this is...
42. ### Continuous functions

Homework Statement Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and $$f:[a,b]\times [c,d]$$ is continuous. Homework Equations The Attempt at a Solution...
43. ### Moment of inertia avoid double integral?

Homework Statement Determine the moment of inertia of the shaded area about the x-axis. Homework Equations I(x)= y^2dA The Attempt at a Solution In order to determine the moment of inertia of the shaded area about the x-axis I first looked at the portion above the x-axis, integrate it with...
44. ### I Integration of 1 variable in 2 different ways.

I have to do a integration which goes like this: (V-M)(dP/dx)+3P(dV/dx)=0, (where M,P and V are constants). If you integrate with dx, you will get: ∫[(V-M)dP]+∫[3PdV]=0. which ultimately results in the answer M=4V. Now, i can put the first equation in this form also...
45. ### Applying the divergence theorem to find total surface charge

Homework Statement Sorry- I've figured it out, but I am afraid I don't know how to delete the thread. Thank you though :) Homework Equations Below The Attempt at a Solution Photo below- I promise its coming! I've started by using cylindrical coordinates, but I wasn't sure if spherical...
46. ### Integration of Maxwell speed distribution function

Homework Statement Show the steps needed to obtain the equation for average molecular speed, cavg=√8RT/πM from the integral (from negative infinity to infinity) ∫v*f(v)dv where f(v) is the Maxwell distribution of speeds function f(v)=4π*(M/2πRT)1.5v2e-Mv2/2RT M is the molar mass of the...
47. ### Trig Sub Integration

1. The problem is as follows: ∫(√1+x^2)dx/(x) 2. Using trig sub --> x = atanΘ with a = √1 = 1. So x = tanΘ and dx = sec^2ΘdΘ. 3. Picture included of attempted solution. I tried u substitution with both u = secΘ and u=tanΘ but didn't have the...
48. ### Surface area of a spherical cap by integration

hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...
49. ### Unusual Limit involving e

This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved \int_{0}^{\infty}x e^{-x}dx=1. Well, when we do Integration by parts we will have let u = x du = dx dv = e^{-x}dx v =...
50. ### Volume of an octagonal dome by using calculus

On this picture we see a octagonal dome. I am trying to calculate the volume of this object by integral calculus but I can't find a way. How would you calculate this? https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202015-12-17%20kl.%2002.14.48.png [Broken] I am majoring in...