Integral Definition and 1000 Threads

I have learned an adequate amount of calculus including implicit, parametric differentiation as well as Upton second order differential equations in high school math course. It was really abstract and we were taught only how to solve mathematical problems. Now, I need to model those problems in...

Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...

I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...

Calculate the following definite trigonometric integral: \[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\].

$\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...

Hi, I have homework question that I'm trying to solve. But I can't understand the basis. Here is a picture of the question and what I have done: My question is, How do I set the boundaries for the integral? 1. If I want the whole squart. 2. To sum up areas. Assume I want to sum the areas, I...

Hello! I am reading a derivation of the path formulation of QM and I am a bit confused. They first find a formula for the propagation between 2 points for an infinitesimal time ##\epsilon##. Then, they take a time interval T (not infinitesimal) and define ##\epsilon=\frac{T}{n}##. Then they sum...

Charles Link submitted a new PF Insights post An Integral Result from Parseval's Theorem Continue reading the Original PF Insights Post.

Hi! I came across a proof in my physics textbook (amperage=wattage/area), and it contained this integration: ∫0 T sin2(ωt) dt The whole thing: 1/T∫0 T sin2(ωt) dt = 1/T(t/2 + sin2ωt/2ω)|T 0 = 1/2 I didn't remember how to integrate that, so I went back to check my notes, and look at it at...

Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...

Hi, I am using Mathematica to calculate density of states and current of the Green's function times self energy in most simple form. I am not sure if I am getting current integral over energy implemented correctly. Shouldnt first current plot be a line with a slope? Below is my code...

How to find the value of miits integral for calculations of the propagation of normal zonei in superconductors?

\tiny{206.08.08.10 } \begin{align*} \displaystyle I&= \int_{-\infty}^{0}\frac{dx}{(x+2)^{1/3}}\\ &=-\infty\\ \end{align*} why does this go to $-\infty$

I am working on the integral representation of the Euler-Mascheroni constant and I can't seem to understand why the first of the two integrals is (1-exp(-u))lnu instead of just exp(-u)lnu. It is integrated over the interval from 1 to 0, as opposed to the second integral exp(-u)lnu which is...

Hello! If I have a real integral between ##-\infty## and ##+\infty## and the function to be integrated is holomorphic in the whole complex plane except for a finite number of points on the real line does it matter how I make the path around the poles on the real line? I.e. if I integrate on the...

I recently came across a problem in Irodov which dealt with the gravitational field strength of a sphere. Took some time to get my head around it and figure how to frame a triple integral, but it felt good at the end. Am I going to start seeing triple integrals in the freshman year tho? If so...

Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##, $$\frac{\partial}{\partial r_k}\int_V...

Hello everyone. Iam trying to get my head around a solution for an integral but I can't figure out how its done. I have given the following : x1'(t) = 0 x2'(t) =tx1(t) Where " ' " indicates the derivative. Talking the time integral the result is given by: x1(t) = x1(t0) x2(t) =...

Homework Statement Find a continuous funciton ##f## such that $$ f(x) = 1+ \dfrac{1}{x} \int_{1}^{x} f(t)dt $$ I think I solved it but I would like to see if it's right. Well, first of all, by the fundamental theorem of calculus I know that $$ \left( \int_{1}^{x} f(t)dt \right) ' = f(x) $$...

In the image below, why is the third line not \frac {ln(cosx)} {sinx}+c ? Wouldn't dividing by sinx be necessary to cancel out the extra -sinx that you get when taking the derivative of ln(cosx)? Also, wouldn't the negatives cancel?

I understand that if you have a function in which you want to determine the full (i.e. account for positive and negative values) integral you need to break down your limits into separate intervals accordingly. Is there any way in which you can avoid this or is it mathematically impossible? If...

Hello! I found a proof in my physics books and at a step it says that: ##\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx \sim_{t \to \infty} e^{-imt}##. Any advice on how to prove this?

I was wondering if anyone knows how to set up a procedure in REDUZE that will decompose tensor integrals appearing in QCD loop calculations into a sum of scalar topologies with the tensor structure factored out? I've had a look at the appropriate manual but I am not entirely sure how to...

Homework Statement Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol 10.9 Exercise 2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the...

So I'm studying a course on measure theory and we've learned that the Lebesgue integral of a real function is (loosely) defined as the total area over the x-axis minus the total area under the x-axis. This seems to me to be limited because these areas can both be infinite but their difference...

1. Homework Statement Guys I am struggling with question 2,4,5 I had upload the question and my attempt. I had been re doing for several times with same answer not match with the model answer, please show me the correct way of solving it Homework EquationsThe Attempt at a Solution

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...

Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...

I am assuming that this line integral is along the straight line from $\displaystyle \begin{align*} (0,0,0) \end{align*}$ to $\displaystyle \begin{align*} \left( 5, \frac{1}{2}, \frac{\pi}{2} \right) \end{align*}$, which has equation $\displaystyle \begin{align*} \left( x, y, z \right) = t\left(...

Evaluate the triple integral. Let S S S = triple integral The function given is 2ze^(-x^2) We are integrating over dydxdz. Bounds pertaining to dy: 0 to x Bounds pertaining to dx: 0 to 1 Bounds pertaining to dz: 1 to 4 S S S 2ze^(-x^2) dydxdz S S 2yze^(-x^2) from y = 0 to y = x dxdz S...

Use a triple integral to find the volume of the solid bounded by the graphs of the equations. x = 4 - y^2, z = 0, z = x I need help setting up the triple integral for the volume. I will do the rest.

Homework Statement A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0 Homework Equations Find a value of a(in...

I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue. The theory is: where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2

So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...

Hi all! I'm new to Mathematica. I have written a code for performing a convolution integral (as follows) but it seems to be giving out error messages: My code is: a[x_?NumericQ] := PDF[NormalDistribution[40, 2], x] b[k_?NumericQ, x_?NumericQ] := 0.0026*Sin[1.27*k/x]^2 c[k_?NumericQ...

Can anyone please tell me significance of these corollaries of fundamental integral theorems? I can prove these corollaries but I don't understand why do we need to learn it? Do these corollaries have some physical significance? (a)$$\iiint_V(\nabla T)d^3 x=\oint_S T d\vec a$$ here S is the...

Homework Statement I have a question. I need to know the integral dxdydz/(y+z) where x>=0, y>=0, z>=0.Homework Equations It is bounded by x + y + z = 1. The transformations I need to use are x=u(1-v), y=uv(1-w), z=uvw. The Attempt at a Solution y+z = uv. J = uv(v-v^2+uv) So I get the integral...

Homework Statement https://holland.pk/uptow/i4/7d4e50778928226bfdc0e51fb64facfb.jpg Homework Equations improper integral The Attempt at a Solution (attached) Whats wrong with my calculation? I cannot figure it out after hours... Thank you very much!

Homework Statement Solve the surface integral ##\displaystyle \iint_S z^2 \, dS##, where ##S## is the part of the paraboloid ##x=y^2+z^2## given by ##0 \le x \le 1##. Homework EquationsThe Attempt at a Solution First, we make the parametrization ##x=u^2+v^2, \, y=u, \, z = v##, so let...

I was wondering if I could get some pointers on how to at least start on this. In quantum mechanics we are using the WKB approximation, and we end up with a definite integral that looks like this: ∫(1 - a(cosh(x))-2)1/2 dx = ∫(1/cosh(x)) (1 - a(cosh(x))2)1/2 dx where a is a positive constant...

Homework Statement derive maxwell distribution function in case of 1-d and 2-d classical gas Homework EquationsThe Attempt at a Solution [/B] The constant K can be solved from normalization. ##\int_{-∞}^{∞} F(V_x)dV_x = 1## substituting ##F(V_x)=Ke^{+/- kV_x^2}## ##1 = K\int_{-∞}^{∞}...

Evaluate the iterated integral by converting to polar coordinates. Let S S = interated integral symbol S S xy dy dx The inner integral limits are 0 to sqrt{2x - x^2}. The outer integral limits are 0 to 2. Solution: I first decided to rewrite sqrt{2x - x^2} in polar form. So, sqrt{2x -...

Hi. I am revising my Mechanics: Dynamics by reading the Beer 10th edition textbook and Pytel 2nd edition In Pytel pg 358 art. 17.3 the angular momentum about the mass center of a rigid body in general motion is being calculated...

I am trying to do the following integral: $$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)}.$$ Wolfram alpha - http://www.wolframalpha.com/input/?i=integrate+(cos(x))^(1/2)+dx+from+x=pi+to+3pi gives me $$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)} = 4f E(2) = 2.39628f + 2.39628if,$$ where E is the...

Homework Statement Homework Equations $$F(x)=\int_a^x f(x),~~F'(x)=f(x)$$ The Attempt at a Solution In F'(x), x is at the end of the domain a-x, so, in my function ##~\cos(x^2)~## i also have to take the end of the domain, and it's 2x, so F'(x)=cos(4x2), but it's not enough. The answer is...

Homework Statement I'm fine with the first part. Part b) is causing me trouble http://imgur.com/xA9CG5G Homework EquationsThe Attempt at a Solution I tried subbing in the solution y1 into the given equation, but I'm not sure how to differentiate this, i thought of using integration by parts...

Homework Statement let's use this symbol to denote the unit impulse function δ When integrating the unit impulse function (from negative infinity to infinity) ∫δ(t) dt I know that this results in a value of 1 and is only nonzero at the point t = 0. However for example take this integral into...

Homework Statement Why specifically 1/2 is the coefficient in CK? the sum, basically, doesn't change except for the coefficient. i can choose it as i want. I understand the sum must equal the integral but i guess that's not the reason Homework Equations Area under a curve as a sum...

Homework Statement $$\int_{-\infty}^\infty \space \frac{cos(2x)}{x-3i}dx$$ Homework EquationsThe Attempt at a Solution $$\int_{-R}^R \space \frac{e^{2ix}}{x-3i}dx + \int_{C_R} \space \frac{e^{2iz}}{z-3i}dz = 2\pi i\sum\space res \space f(z)$$ Then using Jordan's Lemma, as ##R\to\infty## the...