Integrals Definition and 1000 Threads

The following 3 pages are extract from the book: "CALCULUS VOL II" by Tom M. Apostol My interpretation of these 3 pages is worked out in the attached PDF file. Entirely done in propositional logic language. Can anyone point out the mistakes or incorrect logical steps (if any) in the attached...

1#Find the area of the region, enclosed by: 2#Find the area of the region bounded by: 3#in the region limited by: find the solid volume of revolution that is generated by rotating that region about the x axis

where the region of integration is the cube [0,1]x[0,1]x[0,1] my question is where can we use the polar coordinate? is it only usable if the region of integration looks like a circle regardless of the function inside the integral? (if yes it means that using this kind of transformation is wrong...

I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...

$ \displaystyle I = \int_0^{\pi/2} \sin^{\sqrt{2}+1}{x}$ and $\displaystyle J = \int_0^{\pi/2} \sin^{\sqrt{2}-1}{x}$. Find $\displaystyle \frac{I}{J}.$

Please help to see whether it's correct to do in this way

Are there any useful references or resources that intuitively show how Jacobi Elliptic functions [sn, cn, dn, etc] are geometrically interpreted from properties of ellipses? And how the Jacobi Elliptic functions and integrals can be shown to be generalizations of circular trig functions? Thanks!

Archimedes Riemann integral is one of the most elegant achievements in mathematics, I have a great admiration for it. Mr. Patrick Fitzpatrick commented on it as Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer...

Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...

We're given a function which is defined as : $$ f:[0,1] \mapsto \mathbb R\\ f(x)= \begin{cases} x& \text{if x is rational} \\ 0 & \text{if x is irrational} \\ \end{cases} $$ Let ##M_i = sup \{f(x) : x \in [x_{i-1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 ...

Does $$\int_{t=0}^{\infty}f(t)dt=\int_{t=0}^{\infty}g(t)dt$$ imply $$f(t)=g(t)$$ ?

I recently stumbled upon Gradshteyn, Ryzhik: Table of Integrals, Series, and Products and it is worth recommending for all who have to deal with actual solutions, i.e. especially engineers, physicists and all who are confronted with calculating integrals, series and products.

Integrals are defined with the help of upper and lower sums, and more number of points in a partition of a given interval (on which we are integrating) ensure a lower upper sum and a higher lower sum. Keeping in mind these two things, I find the following definition easy to digest A function...

Mentor note: Moved from technical section, so is missing the homework template. Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?

Hi PF! I am trying to computer a matrix of integrals. Think of it something like this: Table[Integrate[x^(i*j), {x, 0, 1}], {i, 0, 5}, {j, 0, 5}] I have 16 cores, and would like to have each core handle a specified amount of integrals. Anyone know how to do this? Thanks so much!

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...

Purcell says that taking the surface integral of the magnetic field ##\textbf{B}## over the surfaces ##S_{1}, S_{2}, S_{3},...## below is a good way of finding the average of the volume integral of ##\textbf{B}## in the neighborhood of these surfaces. More specifically, he says in page...

My work is in the following pdf file:

I can find the area of the triangles but can't solve the squares for some reason

The Euler Lagrange equation finds functions ##x_i(t)## which optimizes the definite integral ##\int L(x_i(t),\dot x_i(t))dt## Is there any extensions of this to multiple integrals? How do we optimize ##\int \int \int L(x(t,u,v),\dot x(t,u,v))dtdudv## ? In particular I was curious to try to...

Suppose we have a region R in the x-y plane and divide the region into small rectangles of area dxdy. If the integrand or the limits of integration were to be simplified with the introduction of new variables u and v instead of x and y, how can I supply the area element in the u-v system in the...

I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...

This is from Horatio Nastase "Intro to Quantum Field Theory" book (Cambridge University Press, 2019) , chapter 59. The reader is supposed to massage equation (3) into equation (4) with the help of the given polylogarithm formulas (1) and (2). I do not see at all how that's possible...

I want to understand where the minus 1 in the first line in the RHS term comes from. I assume the little apostrophe means taking a derivative. But the antiderivative of x^(n-1) is (1/n)x^n. Why the -1? thank you

Summary:: Pretty sure they have something to do with path integrals, or what not. But obviously it's hard to *search* for this stuff. Basically, I'm looking for a textbook, any textbook--physics, mathematics, etc.--that deals with integrals that look something like this (mistakes are mine): S...

If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...

How to find the centroid of circle whose surface-density varies as the nth power of the distance from a point O on the circumference. Also it's moments of inertia about the diameter through O. I'm getting x'=2a(n-2)/(n+2) And about diameter -4(a×a)M{something}

Hi guys, for a project I had to get involved with discrete Fourier transforms to solve PDEs. However, the code that I implemented according to a pseudo-code from a paper did not work - it seems like I calculated integrals incorrectly. To search the error, I tried to integrate the sin(x)...

ok this is a snip from stewards v8 15.6 ex hopefully to do all 3 here $\displaystyle\int_0^1\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}\dfrac{z}{y+1} \,dxdzdy$ so going from the center out but there is no x in the integrand $\displaystyle\int_0^{\sqrt{1 - x^2}} \dfrac{z}{y + 1}dx =\dfrac{ \sqrt{1 -...

15.6.4 Evaluate the iterated integral $$\int_0^1\int_y^{2y}\int_0^{x+y} 6xy\, dy\, dx\, dz$$OK this is an even problem # so no book answer but already ? by the xy

Hi I study optics and many times i found a term called (Overlap integral ) as attached pictures .. I can't understand from where these expression comes (mathematically) and what these functions means in particular ( even from mathematical point of view) I can't understand the nature of...

Summary: I'm looking for a table of Coulomb and exchange integrals for Lithium and beyond. I'm looking for a convenient table of Coulomb and exchange integrals for Lithium and beyond. I've looked everywhere and I find integrals for J,K 1s, 2s or 1s,2p for Helium. Does anyone know of a...

Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. Consider the following multiple integral: ##\displaystyle A=\iiint_{V'} \left[ \iint_S \dfrac{\cos(\hat{R},\hat{n})}{R^2} dS \right] \rho'\ dV' =4 \pi\ m_s## where...

ok this is from my overleaf doc so too many custorm macros to just paste in code but I think its ok,,, not sure about all details. appreciate comments... I got ? somewhat on b and x and u being used in the right places

Richard Feynman formulated quantum path integrals to show that a single photon can theoretically travel infinitely many different paths from one point to another. The shortest path, minimizing the Lagrangian, is the one most often traveled. But certainly other paths can be taken. Using single...

Hi folks, I love doing integrals, and I think I'm going to start a competition at my school. The integrals in the standard calculus textbooks I have access to, Briggs, Stewart, etc., are pretty elementary. I am looking for some harder integrals. I have the books Irresistible Integrals and...

So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector? In particular I’m...

I expanded the exponential with the derivative to get: ## Z = \Bigg(1 + \frac{1}{2} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} + \frac{1}{4} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl}...

We're trying to reduce the tensor integral ##\int {\frac{{{d^4}k}}{{{{\left( {2\pi } \right)}^4}}}} \frac{{{k^\mu }{k^\nu }}}{{{{\left( {{k^2} - {\Delta ^2}} \right)}^n}}}{\rm{ }}## to a scalar integral (where ##{{\Delta ^2}}## is a scalar). We're told that the tensor integral is proportional...

Wave optics, including diffraction, seems to be apt for path integral language. In fact, Feynman's double slit language is purely "diffraction". Also, the PDE for the wave equation results in a solution via Green's function, and the Green function is where "the path integral lives". I have...

1. ##\sum_{n=1}^N \arctan{(n)} \geq N \arctan{(N)}-(1/2)\ln{(1+N^2)} \iff \sum_{n=1}^N \arctan{(n)} \geq N \int_0^N \frac{1}{1+x^2} dx - \int_0^N \frac{x}{1+x^2} dx## Where do I go from here? I've tried understanding this graphically, but to no avail. 2. Maybe this follows from finding an...

Good evening! Going through a bunch of calculations in Ashcroft's and Mermin's Solid State Physics, I have come across either an error on their part or a missunderstanding on my part. Suppose we have a concatenated function, say the fermi function ##f(\epsilon)## that goes from R to R. We know...

Hello all, This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as: $$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \...

For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.

Continue reading...

I was reading about this, and they seem the same. Of course, if they were the same, they wouldn't have different names.

Hi, I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that $$ \int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2), $$ where ##e_{ij}(\hat{q})## is any...