Integration Definition and 1000 Threads

May I ask if the following process is correct? Given: F=ma Apply an impulsive force using the dirac delta near 0 (with F nearly constant over the tiny impulsive interval) ma = Fδ(t) This is a second order differential equation with a forcing function. However, I cannot readily integrate...

Greetings Dear community! Here is the solutions using two different methods: the first method is the Green theorem and the second is the simple path integration method: My question is why they integrate over [0.2pi] in the path integration method while they integrate within [0. pi] in the...

Angle theta is different for every place at airfoil surface, so it can't be one theta from leading edge to trailing edge. Can please someone explain pressure integration in depth, step by step?

$$\int_{0}^{2\pi } (1+2cost)^{n}cos(nt) dt$$ $$e^{it} = z, izdt = dz$$ $$\oint (1+e^{it}+e^{-it})^{n}\frac{e^{nit}+e^{-nit}}{2} \frac{dz}{iz} = \oint (1+z+z^{-1})^{n}\frac{z^{n}+z^{-n}}{2} \frac{dz}{iz}$$ $$\oint (z+z^{2}+1)^{n}\frac{z^{2n}+1}{z^{2n+1}} \frac{dz}{2i} = \pi Res = \pi...

I am trying to perform a four dimensional integration in cpp using gsl. I do this by nesting together the one-dimensional integration routines from gsl library. What I wrote seems to work for a few test integrands but I am having trouble with the integrand that I actually want a value for. See...

(e-√x)/√x (integral from title) I integrated by substituting and the bounds changed with inf changing to -inf and 1 changing to -1 My final integrated answer is -2lim[e-√x]. What happens to this equation at -inf and -1? As I can't put them into the roots

In my job, I was given the task of calculating a force that operates an ultrasound transmitter on a receiver. The calculation is made by assuming that each point on the transmitter is a small transmitter and integration should be made on the surface of the transmitter. Since the transmitter is...

I am interested in particular in the second integral, in the ##\hat{r}## direction. Here is my depiction of the problem: As far as I can tell, due to the symmetry of the problem, this integral should be zero. $$\int_0^R \frac{r^2}{(x^2+r^2)^{3/2}}dr\hat{r}$$ I don't believe I need to...

https://www.physicsforums.com/threa...f-a-translating-and-rotating-pancake.1005990/ So,I think I posted this in the wrong place. So, I will move it to here. Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant. Hence, isn't...

Can someone please tell me the book that contain integration using hyperbolic substitution for beginner? I know that hyperbolic functions is taught in Calculus book but most of them is only some identities and inverses of hyperbolic functions.

Thought it could be fun to have a sort of "PF Integral Bee"... if you know some interesting/quirky/etc. integrals then post them here! 🤓 To get the ball rolling... 1. ##\displaystyle{\int_0^1} \dfrac{\ln{(x+1)}}{x^2+1} dx##

Can someone please show me an example of integration using hyperbolic substitution? Thank you

I know that gradient is multi-variable derivatives. But, here line integration (one dimensional integral) had canceled gradient. How?

$$x(t)=\int \dot{x}(t)\mathrm dt=vt+c$$ That's what I did. But, book says $$x(t)=\int \dot{x}(t)\mathrm dt=x_0+v_0 t+ \frac{F_0}{2m}t^2$$ Seems like, $$x_0 + \dfrac{a_0}{2}t^2$$ is constant. How to find constant is equal to what?

My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!

I realized I never actually derived the kinematic equations of motion for the exact Newtonian gravitational force. For an object falling near the surface of the earth, how do we handle integrating the equation of motion to derive the kinematics equations without using the approximation of...

I'm going to type out my LaTeX solution later on. But in the meantime, can anyone check my work? I know it's sloppy, disorganized, and skips far more steps than I care to count, but I'd very much appreciate it. I'm not getting the answer as given in the book. I think I failed this time because I...

If I have a force that behaves according to the formula ##F(x)=\alpha x-\beta x^3##, how can I get the potential energy from it? I know that: $$-\frac{\mathrm{d}V(x)}{\mathrm{d}x}=F(x),$$ but what about the limits of the integration?

I’ve always struggled with integration and I don’t know how to do this question, I’m not sure what I’m being asked to calculate. I tried to calculate this as a definite integral but there is no boundary conditions for the distance the object has traveled which is confusing any help would be...

Greetings While solving the following exercice, ( the method used is the integration by filaments and I have no problem doing it this way) here is the solution My question is the following: I want to do the integration by strate and here is my proposition is that even correct? I would like...

Greetings! As mentionned my aim is to change the order of integral, and I totally agree with the solution I just have one question: as you can see they have put 0<=y<=1 and 0<=x<=y^2 but would it be wrong if I put 0<=y<=1 and y^2<=x<=1? Thank you!

I'm trying to pass through some parameters of a function to the gsl integration routine but my code is currently not returning correct values. I attach a version of my code using dummy example functions and names. struct myStruct_t { double a; }; double func(double z, void* params)...

I simply plugged \phi = \phi_0 (\eta) + \delta \phi (\eta, \vec x) into the given action to get \begin{align} S &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi^2 -(\nabla \phi)^2\right)-a^4V(\phi) \right] \nonumber \\ &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi_0^2 + (\delta...

First, I calculated the inverse of ##y=e^x## since we're talking about y-axis rotations, which is of course ##x=lny##. Then, helping myself out with a drawing, I concluded that the total volume of the solid must've been: $$V=\pi\int_{0}^{1}1^2 \ dy \ +(\pi\int_{1}^{e}1^2 \ dy \ - \pi...

I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity. Tong's notes and some others are good on the abstract/theoretical side but it'd really be better at this stage to get some practice with concrete examples in order to...

There's an integral over a 2-sphere ##S## with unit normal ##N^a## within a hypersurface orthogonal to a Killing field ##\xi^a##$$F = \int_S N^b (\xi^a / V) \nabla_a \xi_b dA = \frac{1}{2} \int_S N^{ab} \nabla_a \xi_b dA, \quad N^{ab} := 2V^{-1} \xi^{[a} N^{b]}$$which follows because the Killing...

I began this solution by assuming a = x+iy since a is a complex number. So I wrote expressions of <a| and |a> in which |n><n| = I. I got the following integral: Σ 1/πn! ∫∫ dx dy exp[-(x^2 + y^2)] (x^2 + y^2)^n I I tried solving it using Integration by Parts but got stuck in the (x^2 + y^2)^n...

Split the integral $$\frac{Aa}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{ikx}dk - \frac{A}{\sqrt{2\pi}}\int^{\infty}_{-\infty}|k|e^{ikx}dk$$ Apply the boundary conditions, this is where my biggest source of uncertainty comes from I doubled the integral and integrated from 0 to a instead of from -a...

Hi I'm having troubles with integration specially by substitution, I'm going to read a calculus textbook and i need recommendations of books with a good treatment on the different techniques of integration. I'd like a book with good exercises for self study and a exposure to integration of...

Is it possible to do the integration? That is the full question I don't know where to start, try to use ##u=\cos x## and also ##\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)## but failed. Thanks

Good day ! I have a problem with the solution of the floowing integrals Indeed i don't understand why they choose such borders for integral b/a<c y<c doesn't mean that y<b/a ! many thanks in advance!

As we all know, integration by parts can be defined as follows: $$\int u dv = uv - \int v du$$ And the usual strategy for solving problems of these types is to intelligently define ##u## and ##dv## such that the RHS integral can easily be evaluated. However, something that is never addressed is...

This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...

In Griffith’s section 10.3.1, when proving why there is an extra factor in integrating over the charge density when it depends on the retarded time, he makes the argument that there can only ever be one point along the trajectory of the particle that “communicates” with the field point. Because...

∫tan^2 x ( -infinity to +infinity)

I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof. He is evaluating an integral along a contour which makes up the imaginary axis. He has the following: $$ -i\int_{-iR}^{iR}...

I can calculate the value of the integration, it will be ##\frac{\sqrt{3}}{2}## But if I draw the function and consider the area bounded by the curve and x-axis from x = 0.5 to x = 1, it seems that the area will be infinite because x = 1 is vertical asymptote. Why can't I consider from "area...

This is my first attempt ...

my thinking was to have everything changed to a function that has cosine only... ##\int_0^{0.5π} \frac {1-cos^2x}{sin x + cos x}dx## ##\int_0^{0.5π} \frac {(1-cos x)(1+cos x)}{(1-cos^2x)^{0.5} + cos x}dx## ... first of all is this integration possible? if so then let me know if i am on the...

So I am confused about a proof in which the formula for expected value of velocity, ##\frac{d\langle x \rangle}{dt} ##, is derived. Firstly, because the expected value of the position of wave function is $$\langle x \rangle =\int_{-\infty}^{+\infty} x|\Psi(x,t)|^2 dx$$Therefore...

Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...

Homework Statement:: The magnetic field at every point on the path of integration Relevant Equations:: The scenarios/situations are shown in the attached photo. "Any conductors present that are not enclosed by a particular path may still contribute to the value of B field at every point, but...

Details of Question: ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into: s − s0 = v0t + ½at2 My main question is about the integration of...

I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example: Let ##M## be a smooth...

Anyone have any idea how to perform the following two integrals? ##\int d\Omega n_{i}n_{j}## and ##\int d\Omega n_{i}n_{j}n_{k}n_{l}## where the n is a unit vector.

I think in the case of "n da" you can see the denominator (1+x^2) as a constant, so ∫ ( sin(a) + M^2 ) / ( 1 + x^2 ) da = ( 1 / ( 1 + x^2 ) ) * ∫ (sin(a) + M^2 ) da = ( 1 / ( 1 + x^2 ) ) * ( -cos(a) + (M^2)a ) = ( - cos(a) + (M^2)a ) / ( 1 + x^2 ) --- Is this the way to go? This is my...

Want to integrate the total energy density over all photon energies between two temperature values from 500K to 5800K, but not sure how to proceed. Here is some examples to help:

Can you help me with this question ? I am really struck with this question. Thank you in advance.