Integral Definition and 1000 Threads

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 )]...

I got the following two integral for the a particular solution of a 2nd order linear ODE $$(D-a)(D-b)y = g(x)$$ by using inverse operators ##\frac{1}{D-a}## and ##\frac{1}{D-b}##. The two different integrals are obtained by operating these operators in different order on y to get a particular...

1. The problem statement, all variables, and given/known data Find and categorize extremes of the following function: $$F(y)=\int_{y}^{y^{2}}\frac{1}{\ln^{2}x}dx$$ for ##y>1##. Homework Equations $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...

Dear all, Please solve this integral: I tried integral by substitution, but failed. Wolframalpha shows the result is 6, but I don't know how to proceed it. Can it be solved by elementary function?

Hi. So I'm trying to use Laplace transforms in inverting a particular s-function via the convolution formula. I ended up with this terrifying-looking thing: So distributing, I ended up with: Evaluating the second integral poses no problem for me (although I think the integration will...

Evaluate What I tried so far is to break the denominator as (1+Cos2x). The integral of 1/(x^2+2) can be done with substituting x = sqrt(2)u and will evaluate to a constant times arctan (x/sqrt(2)) but I have no idea how to evaluate the rest. This is calculus AP (with real numbers only). My...

Solve the definite integral \[I(a) = \int_{0}^{2\pi}\frac{\sin^2 \theta }{1-2a\cos \theta + a^2}\: \: d\theta,\;\;\; 0<a<1.\]

Hey! :o I want to calculate $\iint_{\Sigma}(x^2+y^2)zdA$ on the part of the plane with equation $z=4+x+y$ that is inside the cylinder with equation $x^2+y^2=4$. We can define the surface $\Sigma : D\rightarrow \mathbb{R}^3$ with $\Sigma (x,y)=(x,y,4+x+y)$, where $D$ is the space that is...

Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...

When working a proof, I reached an expression similar to this: $$\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$$ I've tried the following: 1. I tried squaring and combining and converting to polar coordinates, like one would solve a standard Gaussian. However...

Homework Statement A block of mass ##m = 1.00 kg## is being dragged through some viscous fluid by an external force ##F = 10.0 N##. The resistive force can be written as ##R = -bv##, where ##v## is the speed and ##b = 4.00 kg/s## is a phenomenological constant. You may ignore gravity (we...

Homework Statement z=x^2+xy ,y=3x-x^2,y=x find the volume of the region Homework EquationsThe Attempt at a Solution I graphed y=3x-x^2 and y=x I am confused on which region I use to find the volume. Do I use the upper region or the lower region.

Is the "function" R->R f(x) = +oo, if x =0 (*) 0, if x =/= 0 Lebesgue measureable? Does its Lebesgue Integral exist? If yes, how much is it? (*) Certainly we shoud give a convenient meaning to that writing. -- lightarrow

I'm trying to fit a model curve to some data by performing a chi square minimisation wrt three parameters a,b and NN. The trouble I am having is that the variables with which I want to minimise the chi square with respect to appear in an integral. I attach the code I am working with...

I'm reading "Teaching Feynman’s sum-over-paths quantum theory" by Taylor et al. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.374.4480&rep=rep1&type=pdf, I'd like to confirm whether my understanding is correct, so a couple of questions. 1. We need to try and think of all kinds of...

Equation \varphi(x)=x+1-\int^{x}_0 \varphi(y)dy If I start from ##\varphi_0(x)=1## or ##\varphi_0(x)=x+1## I will get solution of this equation using Picard method in following way \varphi_1(x)=x+1-\int^{x}_0 \varphi_0(y)dy \varphi_2(x)=x+1-\int^{x}_0 \varphi_1(y)dy \varphi_3(x)=x+1-\int^{x}_0...

Hi guys, So I'm trying to compute this Feynman integral: $$ V=\dfrac {-i} {2} \int {\dfrac {d^4 k} {(2\pi)^4}} \dfrac {1} {k^2 - m^2} \dfrac {1} {(k+P_1)^2 -m^2} \dfrac {1} {(k+P_1 +P_2)^2 -m^2}$$ I have introduced the Feynman parameters and now have the integral: $$ V=-i \int...

Homework Statement Hydrostatic force on a plane surface ex: Hydrostatic force on a gate: Homework Equations The Attempt at a Solution why can't we just use the formula in the red box above for problem 3.57? instead I have to use integral I am confused, how does this gate different from...

$\tiny{t1.11}$ $\textsf{Evaluate the Integral}$ \begin{align*}\displaystyle I_{11}&=\int \frac{\sin\sqrt{t}}{\sqrt{t\cos^3\sqrt{t}}}\, dt\\ &=\int\frac{\sin\sqrt{t}}{\sqrt{t}\cos^{3/2}\sqrt{t}}\, dt\\ u&=\cos\sqrt{t}\\...

Homework Statement Calculate ##\int x^2 dV## over an ellipsoid with semi-axes a, b and c along x, y and z. rotating around the z axis with an angular speed ##\omega##. Homework EquationsThe Attempt at a Solution I managed to calculate this in the case when it is not rotating and I got...

Homework Statement I'm given the integral show in the adjunct picture, in the same one there is my attempt at a solution. Homework Equations x = r.cos(Θ) y = r.sin(Θ) dA = r.dr.dΘ The Attempt at a Solution [/B] I tried to do it in polar coordinates, so I substituted x=r.cos(Θ) y=r.sin(Θ) in...

$\tiny{2214.t1.14}$ $\text{Evaluate the Integral:}$ \begin{align*}\displaystyle I_{14}&=\int \frac{12\tan^2x \sec^2 x}{(4+\tan^3x)^2} \, dx \\ \textit{Use U substitution}&\\ u&=4+\tan^3x\\ \, \therefore dx& =\dfrac{1}{3\sec^2\left(x\right)\tan^2\left(x\right)}\,du\\ &=4...

So Feynman's path integral considers every possible path that a particle could take from start to end. In that process, there would be a path which contains a segment from, say, A to B at time t. But there could also be a path with a segment from B to A at that same time, t. If so, would this...

Homework Statement Use Green's Theorem to find the area of the region between the x-axis and the curve parameterized by r(t)=<t-sin(t), 1-cos(t)>, 0 <= t <= 2pi Attached is a figure pertaining to the question Homework Equations [/B] The Attempt at a Solution Using the parameterized...

Homework Statement Problem 1: Use double integrals to find the volume of the solid obtained by the rotation of the region: ##\triangle = \left\{ (x, y, z) | x^2 \le z \le 6 - x, 0 \le x \le 2, y = 0 \right\} ## (edit) in the xz-plane about the z axis Homework Equations Volume = ##\int_a^b...

I have the following expression : $$ y_{E} = \int_{0}^{\infty} 0.5 * [E_{1}(µ(E)*r) - E_{1}(\frac{µ(E)*r}{cos \alpha})] * f(r) dr $$ where : - $y_{E}$ has been measured for some E (something like 5 different $E_{i}$, to give you an idea) - µ(E) is retrieved from a table in the litterature...

Hi, todos: Do you know how to calculate the definte integral for Integral for ##\int_{-1}^{1} [P_{l}^{m}]^2 \ln [P_{l}^{m}]^2 dx##, where ##P_{l}^{m} (x)## is associated Legendre functions. Thanks for your time and help.

Homework Statement Use double integrals to find the areas of the region bounded by ##x = 2 - y^2## and ##x = y^2## Homework Equations Volume = ##\int_a^b \int_{f(x)}^{g(x)} h(x) dx dy##.. and this is equivalent if I switched the integrals and redid the limits of integration The Attempt at a...

I am going through Shankar's treatment of Feynman Integrals right now, and I have one lingering doubt that I can't quite seem to work out. I was pretty happy with the idea of discretizing time, then doing independent sums over xi at each time. But Shankar simply says that we can consider the...

Homework Statement part b of below [/B] Homework Equations ##(1+x)^{1/2}=1+\frac{1}{2}x-\frac{x^{2}}{8}+...## The Attempt at a Solution [/B] ##\int\limits^{\Lambda}_{-\Lambda} \frac{dy}{\sqrt{r^2+y^2}}=log(\lambda+\sqrt{\lambda^2+r^2}) - log(-\lambda+\sqrt{\lambda^2+r^2}) ## ##=...

Homework Statement I am supposed to evaluate the contour integral of the positive branch of ##z^{-1/2}## over the following contour: I believe the answer should be 0, by Cauchy's theorem (loop encloses no poles), but my methods of parameterization have led to non-zero answers. Homework...

Suppose we have a function $f(x)$ which is infinitely differentiable in the neighborhood of $x_0$, and that: $f^{(k)}(x) \ge 0$ for each $k=0,1,2,3,\ldots$ for all $x$ in this neighborhood. Let $R_n(x)=\frac{1}{n!}\int_a^x f^{(n+1)}(t)(x-t)^n dt$ where $x_0-\epsilon <a<x<b<x_0+\epsilon$; I...

Homework Statement the second solution is the correct, I know you can put C on both sides and it simplifed to C2 on one side, but why can't you put C2 on the right side? Homework EquationsThe Attempt at a Solution

Homework Statement Find the z -coordinate of the center of mass of the first octant of a sphere of radius R centered at the origin. Assume that the sphere has a uniform density. Homework Equations Mass = Integral of the density function Center of mass for z = Integral of density * z divided...

I know this was done before on this forum but can't find it $\displaystyle \int\sqrt{x^2+1}\ dx%$ $\text{where $u=x$ and $a=1$ then plug}$ $\displaystyle \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\ln\left| u+\sqrt{u^2+a^2}\right|$ how is this derived?

Hey! :o Let $D$ be the space $\{x,y,z)\mid z\geq 0, x^2+y^2\leq 1, x^2+y^2+z^2\leq 4\}$. I want to calculate the integral $\iiint_D x^2\,dx\,dy\,dz$. I have done the following: We have that $x^2+y^2+z^2\leq 4\Rightarrow z^2\leq 4-x^2-y^2 \Rightarrow -\sqrt{4-x^2-y^2}\leq z\leq...

Hi, Could you please help me understand the following example from page 76 of "QFT for the gifted amatur"? I can't see how the following integral becomes Thanks a lot

So for an infinite plane of current, current traveling in the X direction, the magnetic field everywhere above the plane is going clockwise, and the m. field below the plane is going counterclockwise. So the path integral is Integral of H dot dl = Current Enclosed Why, in this video, does the...

\begin{align*}\displaystyle \vec{V_2} &=\int_0^3 \left[\left( \frac{4}{\sqrt{1+t}}\right){I}-\left(7t^2 \right){j} +\left(\frac{14t}{\left(1+t^2 \right)^2}\right){k}\right] dt \\ &=\left[\int_0^3 \frac{4}{\sqrt{1+t}} dt \right] {I} -\left[\int_0^3 7t^2 dt\right] {j} +\left[...

Homework Statement Homework Equations flux = int(b (dot) ds) The Attempt at a Solution I just wanted clarification on finding ds. I understand why ds is in the positive yhat direction (just do rhr) but I don't understand where the dxdz come from. How do we find ds in general?

I used Newtons method and taylor approximations to solve this equation $$f'''+\frac{m+1}{2}ff''+m(1-f^{'2})=0$$ It solves for velocity of air over a flat plate. The velocity is a constant ##u_e## everywhere except in a boundary layer over the plate, where the velocity is a function of distance...

I am trying to find a series representation for the following expression $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant. Is there a name for this function? I found a series representation for $$\int_{i=0}^\infty...

##\Gamma(x)=\int^{\infty}_0 t^{x-1}e^{-t}dt## converge for ##x>0##. But it also converge for negative noninteger values. However many authors do not discuss that. Could you explain how do examine convergence for negative values of ##x##.

Just a quick thought I had: If we have the following integral: ##\int_0^ax^xdx## where ##a>0##, is there any way to tell if all real number results will be transcendental? And if not ##a∈R##, would it be possible if we restrict it to only integers?