Integral Definition and 1000 Threads

Could anyone can tell me how to calculate this type of intergretion. Thanks very much $$\int\frac{{y}^{3}}{(196 - {y}^{2})\times \sqrt{196 - {y}^{2} - {a + y}^{2}}}$$

Hello, For calculating the mean power at a specific cross section of a waveguide, one can calculate the mean value of the temporal function of Poynting Vector, P(t), where P(t) is the ExHy-EyHx. Note that I am not talking about phasors or a sinusoidal state. If I integrate over the waveguide...

Hi, I have the following integral which I am not confident on how to interpret (solve):\begin{equation} \alpha \bigg( \int_0^L [\frac{d^3}{dx^3}\phi] \psi dx - \int_0^L [\frac{d^3}{dx^3}\psi] \phi dx \bigg) \end{equation} at this stage, I am not sure which rule to use to solve each of the two...

Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2} $$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...

$tiny{244.T.15.5.11}$ $\textsf{Evaluate the triple integral}\\$ \begin{align*}\displaystyle I_{\tiny{11}}&=\int_{0}^{\pi/6}\int_{0}^{1}\int_{-2}^{3} y\sin{z} \, d\textbf{x} \, d\textbf{y} \, d\textbf{z}\\ &=\int_{0}^{\pi/6}\int_{0}^{1}...

Evaluate the definite integral $$\int_{0}^{1} \ln^2(1+x^{-1}) \,dx$$

\begin{align*}\displaystyle I_{15.5.8}&=\int_{0}^{\sqrt{2}} \int_{0}^{3y} \int_{x^2+3y^2}^{8-x^2-y^2} dz \ dy \ dx \\ &=\int_{0}^{\sqrt{2}} \int_{0}^{3y} \Biggr|z\Biggr|_{x^2+3y^2}^{8-x^2-y^2}\\ &=\int_{0}^{\sqrt{2}} \int_{0}^{3y} 8-2x^2-4y^2 \ dy \ dx \\ &=\int_{0}^{\sqrt{2}}\Biggr|8y-2x^2...

Homework Statement "Suppose that the accident rate for one workplace ##A## is ##k## times the rate of another workplace ##B##. In other words, ##\lambda_A(t)=k⋅\lambda_B(t)##. Conclude that the probability of no accidents in workplace ##A## is the probability of no accidents in workplace ##B##...

Homework Statement find f(x) which satisfies f(x) = x + ##\frac{1}{\pi}## ##\int_{0}^{\pi} f(t) \sin^2{t} \ d(t)## Homework EquationsThe Attempt at a Solution to solve f(x), I have to solve the integral which contains f(t). And f(t) is the f(x) with variable t? if yes, I will get integral...

I want to prove the following inequality: $$\sum\limits_{k\in\mathbb{N}}\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \big\|f\big\|^2 \sum\limits_{k\in\mathbb{N}}\Big (\int\big|g(x-k)\big|dx\Big)^2$$ where $$\|f\|^2=\int |f(x)|^2dx.$$ My attempt: Just prove the following inequality...

Hello I have a question regarding something we wrote in class today. Let ##A## be a bounded subset of ##R^n##, let ##f,g:A\to \mathbb{R}## be integrable functions on A. ##a)## if ## A## has a volume and ##\forall x \in A :m\leq f(x) \leq M## then ##mV(A)\leq \int_{A}f(x)\leq MV(A)## this...

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.

∫x^2/(4x+1)^10 dx i know that solving this is possible by partial fractions like A/4x+1 + (Bx+C)/(4x+1)^2 + (Dx+E)/(4x+1)^3 and so on but i would like to know if there is another way of solving this.

Homework Statement Your task is to estimate how far an object traveled during the time interval 0≤t≤8, but you only have the following data about the velocity of the object. *First image You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue...

Homework Statement The speed of a runner increased during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. It follows the image's square. Homework Equations...

Homework Statement I have the following integral, $$\frac{1}{\sigma \sqrt{2\pi} t} \int_{-\infty}^{0} \exp[\frac{-1}{2\sigma ^2} (\frac{x-x_0}{t} - p_0)^2]dx$$ that I wish to write in terms of the error function, $$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-g^2}dg$$ However, I can't seem...

A scalar field theory with potential $$V(\phi)=-\mu^2\phi^2+\lambda \phi^4$$ is spontaneously broken and as a consequence, for the ground state, $$\langle \phi(x) \rangle \neq 0$$. However, the path integral, which should give ground state expectation values, looks to be zero by oddness of the...

Homework Statement Homework Equations Using Cauchy Integration Formula If function is analytic throughout the contour, then integration = 0. If function is not analytic at point 'a' inside contour, then integration is 2*3.14*i* fn(a) divide by n! f(a) is numerator. The Attempt at a Solution...

Homework Statement [/B]Homework Equations Substitution. The Attempt at a Solution Since the circle is of unit radius and around origin, limits are x = -1 to 1, and y = -1 to 1 I replaced x by cos t, and y by sin t. But what to put in place of ds? I thought about divergence theorem, but then...

Homework Statement Homework Equations The Attempt at a Solution I'm confused avout questions 2-3. The answers for 2-2 is 1 So the answer for 2-3 is $$\frac{1}{3}$$ But, how the area looks like? Because $$ x^2 $$ will be an open curve upside? There's no boundary for above side.

Hi, I'm trying to calculate an integral which looks unfortunately divergent. The structure is similar to a loop integral but the appendix in the Peskin textbook didn't have a useable form. The integral form is (I did a u substitution to make it easier to look at) \int_x^{\infty}du...

https://1drv.ms/w/s!Aip12L2Kz8zghV6Cnr8jPcRTpqTX https://1drv.ms/w/s!Aip12L2Kz8zghV6Cnr8jPcRTpqTX My question is in the above link

Evaluate \[\int_{0}^{2\pi}\cos x \cos 2x \cos 4x \cdot \cdot \cos 2^{2017}x \cos (2^{2018}-1)x \: dx\]

Homework Statement integration with respect to x Homework Equations integral 1/sqrt (a^2 - x^2) = arcsin(x/a) The Attempt at a Solution image attached, the arcsine term in 5/2 arcsin((2x-5)/5) it should be 5 arcsine(sqrt(x/5))

I have a system of equations, and one of them is this : ##\int(1-U(y))Dy - H*\int(U(y)-U(y)^2)dy=0## Can Newtons method work if I approximate this integral to be ##\sum_y(1-U(y))-H\sum(U(y)-U(y)^2)=0## y is a set integers in range ##[1,1000]## I have Newtons method working for this same system...

Homework Statement Homework EquationsThe Attempt at a Solution Line integral of a curve ## I = \int_{ }^{ } yz dx + \int_{ }^{ } zx dy + \int_{ }^{ } xy dz ## with proper limits. ## I = \int_{\frac { \pi }4}^{ \frac { 3 \pi} 4} abc ( \cos^2 t - \sin^2 t ) dt = -abc ## |I| = abc...

Homework Statement Find the integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2. Homework Equations None. The Attempt at a Solution sin^7 x means sinx to the 7th power. But how do I find this strange integral? I don't think u-substitution, trig identity, any of them will work.

Homework Statement Find the integral of 1/(1+cosx) dx from -pi/2 to pi/2. Homework Equations None. The Attempt at a Solution Here's my work: 1/(1+cosx)=(1-cosx)/((1+cosx)(1-cosx))=(1-cosx)/(1-cos^2 x)=(1-cosx)/sin^2 x This is what I've got so far. But this doesn't seem to simplify the...

Homework Statement Homework Equations This is solution of Griffith problem 11.16 The Attempt at a Solution This is procedure to get a 1-D integral form of Schrödinger equation. I don't understand why that contour integral include only one pole for each contour?

This is rather basic, and may be a misconception of the notation, however, I can't make the following sum up: The following is given: x_n(t) = 1 -nt , (if 0 <= t <= 1/n) and 0, (if 1/n < t <= 1) However, this part I can't grasp this part in the book: \begin{equation} ||x_n||^2 = \int_0^1...

Homework Statement Homework Equations This is a calculation about differential cross section of Yukawa potential. The Attempt at a Solution I can't understand how that highlighted part can be -1 , we don't know if the parenthesis term (iq-1/a) is negative or positive tho.

Prove, that the definite integral $$\int_{0}^{\pi}\ln (2-2\cos x)dx = 0.$$

1) Find the radius of curvature at any point of the cycloid x = a(\theta + sin\theta)y = a(1- cos\theta). 2) Find the radius of curvature at the point (3a/2 , 3a/2) for the curve x3 + y3 = 3axy

Hi, I'm new to this forum. This semester I took Calculus I and just took the final yesterday. There were a few questions that were unexpected that I didn't know how to handle. This integral has got me stumped.$$\int_{0}^{1} e^{x}/(1 + e^{2x}) \,dx$$ The techniques I know at this point...

The problem is to determine the shearforce Q on the hut near the ground. This is not a homework or anything like that, I'm just studying for an exam and this problem is in the book "Engineering Mechanics, Statics" By Meriam Kraige. On another forum, I found this: I understand the part in...

Evaluate $\lim\limits_{{n}\to{\infty}} \int_{n}^{n+1} \cos^2(x^2) \,dx$ I've tried using the half angle identity and the taylor series on the remaining $1/2 + \cos(2x^2)$ to prove the value is $1/2$, but I am out of ideas.

Hi, I have read about the rayleigh sommerfeld integral and its a surface integral. Why is it difficult to calculate the integral directly?

Hi, as you can see at the end of the picture/attached file collision integral is approximated to a discrete sum. Could you express how this approximation is derived?

Homework Statement Evaluate the triple integral y^2z^2dv. Where E is bounded by the paraboloid x=1-y^2-z^2 and the place x=0. Homework Equations x=r^2cos(theta) y=r^2sin(theta) The Attempt at a Solution I understand how to find these three limits, -1 to 1 , -sqrt(1-y^2) to sqrt(1-y^2) , 0 to...

I am having problem on understanding the below solution regarding constant of integration. On integrating an differential equation of RL circuit , for e.g $$10i + 3\frac{di}{dt} = 50 $$ $$i.e \frac{di}{50-10i} =\frac{dt}{3}$$ Integrate $$\frac{1}{10} \int\frac{1}{5-i} di = \frac{1}{3}∫dt...

Hey! :o I want to calculate $\int_{\sigma}\left (-y^3dx+x^3dy-^3dz\right )$ using the fomula of Stokes, when $\sigma$ is the curve that is defined by the relations $x^2+y^2=1$ and $x+y+z=1$. Is the curve not closed? Because we have an integral of the form $\int_{\sigma}$ and not of the form...

Show that $$\int_0^\infty dx\exp(ikx^3) , k>0$$ may be written as integral from 0 to ##\infty## along the line ##arg(z) = \frac{\pi}{6}##. I'd appreciate it if you can help me how to approach this problem. My initial impression was to expand the integrand out...

Homework Statement r=1 and r=1+cos(theta), use a double integral to find the area inside the circle r=1 and outside the cardioid r=1+cos(theta) Homework EquationsThe Attempt at a Solution I am confused on the wording and how to set it up. I tried setting it up by setting theta 0 to pi. and r...

Homework Statement [/B] I would like to ask for Q5b function G & H Homework Equations answer: G: -2pi H: 0 by drawing the vector field The Attempt at a Solution the solution is like: by drawing the vector field, vector field of function G is always tangential to the circle in clockwise...

Homework Statement \int\frac{x^2}{\sqrt{x^2+4}}dx Homework Equations n/a The Attempt at a Solution Letting x=2tan\theta and dx=2sec^2\theta d\theta \int\frac{x^2}{\sqrt{x^2+4}}dx=\int\frac{4tan^2\theta}{\sqrt{4+4tan^2\theta}}2sec^2\theta d\theta=\int\frac{8tan^2\theta...

Hi there, I am having some difficulty evaluating a repeated integral, which is the first of two shown in the image. I had hoped to be able to use Gaussian Quadrature to provide a numerical result, however am unsure on if this is possible for a repeated integral? I have attempted to use Cauchy'...

Homework Statement A massless string of length 2l connects two hockey pucks that lie on frictionless ice. Aconstant horizontal force F is applied to the midpoint of the string, perpendicular to it (see right figure). How much kinetic energy is lost when the pucks collide, assuming they stick...

Hey! :o I want to calculate $$\iint_{\Sigma}\left (ydy\land dz+zdz\land dx+zdx\land dy\right )$$ where $\Sigma$ is the surface that is described by $x^2+y^2+z^2=1$ and $y\geq 0$ and has such an orientation that the perpendicular vectors that implies have a direction away from the point...

Evaluate (use attached figure for depiction) $ \iint_{R} \, xy \, dA $ where $R$ is the region bounded by the line $y = x - 1$ and the parabola $y^2 = 2 x + 6$. I will post solution in just a moment with a reply.

Figure for example 1. ∫∫dydx 2. ∫∫x dydx (x2<= y <= 4) and (-2 <= x <= 2) 1. Two method get the same answer 2. Two method get the different answer - between answers 0 or 8 , the correct answer is ? I met this problem with my problem ∫ (x^4 + y^2)dx +(2x^2-y^4)dy = ∫∫ [(4x - ( 2y )]...