-3
votes
1answer
36 views

How to find the integral of $\int \frac{GMm}{r^2}\,dr$ [on hold]

I want to find the integral of: $$\int_R^\infty \frac{GMm}{r^2}\,dr$$
1
vote
1answer
40 views

Why can we make this integral change of limits? Is it obvious?

When deriving the equation for the impulse-momentum theorem, the following occurs: $$\cdots=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=\cdots$$ I know the ...
4
votes
0answers
36 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
0
votes
0answers
27 views

Properties of functional integration

this question comes from theoretical Physics, the issue being the so called Path Integral. The measure of this thing is something written as $[d\phi]=\prod_x d\phi(x)$ And this should be the limit ...
1
vote
3answers
45 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
1
vote
1answer
43 views

How to properly generalize a definite integral?

I know, I know. On the can, this problem seems simple. Just take $\int_a^bf(x)\mathrm{d}x$ and write is as $\int f(x)\mathrm{d}x$. However, when I tried to do that on an Engineering Dynamics ...
0
votes
2answers
48 views

Finding the average of the absolute value of a function?

I know that to find the absolute value of a function, $$\frac{1}{b-a}\int^b_af(x)\mathrm{d}x$$. This is actually kind of intuitive. The problem is I don't know how to to find the average of the ...
0
votes
2answers
41 views

Calculate Displacement when velocity is a function of displacement

Particle is moving on a straight line and where velocity varies with its displacement as $v=\sqrt{4+4s}$. Find displacement at t = 2 s if s=0 at t=0. I am not able to figure out how to approach this. ...
2
votes
1answer
40 views

Scale-invariance of $\int_0^\infty \frac{f(x)}{x} \ dx$

Let $f$ be some non-negative, measurable function on $[0,\infty)$. The quantity $\int_0^\infty \frac{f(x)}{x} \ dx$ is scale-invariant in the sense that, if one puts $f_c(x) := f(cx)$ for $c > 0$, ...
1
vote
0answers
38 views

Integration in physics and calculations with $dx$

I'm in a physic formation and we are used to play with the infinitesimal elements $dx$ of integration like a variable (for example the calculation of the pressure of a gas), because we look at small ...
0
votes
2answers
39 views

Calculate the energy in a circuit containing a resistor

A voltage peak in a circuit is caused by a current through a resistor. The energy E which is dissipated by the resistor is: Calculate E if Can anyone please give me some suggestions where to ...
5
votes
1answer
73 views

Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
1
vote
0answers
48 views

Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
4
votes
2answers
125 views

What are integrating factors, really?

I can follow the rationale for integrating factors well enough, but they still feel like voodoo to me. Every single description of integrating factors I've seen (and I've seen quite a few, including ...
2
votes
1answer
46 views

Integration problem related to physics problem

I want to know how to solve this (acytually this is the flux on any non-adjacent side of a cube due to a charge q on the vertice of the cube, side length: l) $$a=\oint\vec{E}.\vec{dS}=\oint ...
0
votes
1answer
43 views

Proving the moment of inertia formula for right cylinder

I have a question on whether I can do this with an integral: When I tried solving this, I got (1/2)(M^2)(R^2) instead of (1/2)MR^2 Problem: http://i.imgur.com/QDdEEse.jpg *Sorry for lack of TeX. ...
1
vote
1answer
84 views

How do I solve this integral with hyperbolic functions?

I was studying mechanics when I f ound a problem that lead to an integral that I can't solve. Basically the problem asked to find the period of oscillation function of the energy $E$ of a particle ...
0
votes
0answers
45 views

Integral of P(x)/(A(x)E(x)

I have a question where I have to use the formula $\delta = \int_0^L \frac{P(x)}{A(x)\epsilon(x)}$. It is used to find the 'displacement' or the 'elastic deformation' of an axially loaded member like ...
0
votes
2answers
89 views

Electric charge on a disk

Electric charge is distributed over the disk $x^2+y^2\leq1$ so that the charge density at $(x,y)$ is $\sigma(x,y)=18+x^2+y^2$ coulombs per square meter. How can I find the total charge on the disk? ...
0
votes
1answer
62 views

Heat energy per unit mass necessary to raise temperature in a slice

Suppose that the specific heat is a function of position and temperature $c(x,u)$. show that the heat energy per unit mass necessary to raise the temperature of a thin slice of thickness $\Delta x$ ...
1
vote
1answer
49 views

Line Integral to Find Work on Slope (Without Explicit Use of Vector Calculus Format)

Ok, so my math class has introduced the Line Integral as a way to find the work done on a two-dimensional slope by gravity on an object traversing any distance on the slope. This is all supposed to be ...
0
votes
1answer
89 views

Center of mass in a straight rod

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $$\int_S x_1 \, dx_1 \, dx_2=0 $$ $$\int_S x_2 \, dx_1 \, dx_2=0 $$ ...
2
votes
0answers
30 views

How to calculate the following 3D ${\bf k}$-space integral?

I'm struggling to calculate$$ \sum_{a,b=\pm}\int\frac{\text d\mathbf{k}}{(2\pi)^3} ...
3
votes
2answers
130 views

calculation of Stefan's constant

In the calculation of Stefan's constant one has the integral $$J=\int_0^\infty \frac{x^{3}}{\exp\left(x\right)-1} \, dx$$ which according to Wikipedia is equal to $\frac{\pi^4}{15}$. In this page of ...
0
votes
2answers
102 views

trouble understanding integration

I am reading through this physics book and have trouble understanding how they integrated one of the problems the conditions are Conditions: // ignoring the constant for simplicity $$r = \sqrt{x^2 + ...
1
vote
1answer
327 views

Pumping Water out of Parabolic tank?

First of all, I understand how to do the integration part of this problem, but I am confused about the setup. Here is the question: Use integration to find the work done pumping all the water ...
0
votes
1answer
167 views

Integrals - center of mass of non-uniform density sphere

Given the density in a given spherical co-ordinate $\rho(r,\theta,\phi) = \rho_0 e^{-r/R} (1-cos \theta)$ find the center of mass of the sphere. I managed to get using infinitesimal sized the ...
4
votes
1answer
113 views

Integral evaluation

Evaluate $$\int_{0}^{2\pi}\int_{0}^{\pi} {\cos\phi \sin\phi \over \sqrt{R^2+r^2-2Rr(\cos\phi \cos\theta+\sin\phi \sin\theta \cos\psi )}} d\phi\ d\psi$$ where $R,r,\theta$ are all constants. ...
2
votes
1answer
60 views

Hankel trasformation of acoustic wave equation

We consider a simplified version of acoustic wave equation \begin{equation} \frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^2 p}{\partial z^2}+k^2 ...
0
votes
1answer
36 views

Obtaining $S(t)$ when $a(S)$ is given?

I have the acceleration as a function of distance, $a(t)$ $$a(t) = f(S)$$ $$\int v.dv = \int f(S).dS$$ And so I have velocity as a function of time if I want it. What I need is to find $S(t)$. I ...
1
vote
1answer
1k views

Emptying water out of a Conical Tank? (Calculus)?

Please help me with this Calculus question. I'm not asking you to do the whole thing, but I just need help setting up the height function. Here is the question: A conical tank of radius $6$ feet and ...
0
votes
1answer
48 views

Finding time t for a body with air resistance k to reach to location x

Since gravity for this problem is irrelevant I started from the following equation: $$ma = -kv$$ From here I integrated both sides in order to find an expression of v as a function of t: V stands ...
0
votes
1answer
2k views

A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. (used hooks law this time)

A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length? ok i used ...
0
votes
2answers
578 views

A variable force of 5x^-2 pounds moves an object along a straight line when it is x feet from the origin.

A variable force of $\dfrac{5}{x^2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1$ ft to $x=10$ ft. I tried ...
0
votes
1answer
88 views

Does integration over one complete cycle equals to 4 times integration over quarter-cyle?

From the article pendulum(mathmetics) from wikipedia. There is a demonstration that this equation: $$\dfrac{dt}{d\theta } = ...
2
votes
0answers
183 views

Simple pendulum: Rewriting integral in elliptic integral

I've been reading through the math on nonlinear (large amplitude) pendulum in wikipedia, and it beats me on how can $\displaystyle\int_{0}^{\theta_0} \dfrac{d\theta}{\sqrt{\cos(\theta) - ...
0
votes
1answer
126 views

Surface integral and the divergence theorem

I haven't done a surface integral in a while so I am asking to get this checked. $\mathbf{F} = \langle x, y, z\rangle$ and the surface is $z = xy + 1$ where $0\leq x,y\leq 1$. $\hat{\mathbf{n}} = ...
5
votes
2answers
327 views

Integrating pressure with respect to time

I am trying to work through the math derivation presented in a paper about gas flowing through rock due to a pressure differential across the length of rock. This is my first post so forgive me if I ...
10
votes
3answers
326 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following ...
1
vote
1answer
403 views

Electric field of finite sheet: Full analytical solution of integration?

I am trying to work out the integral $$E_{z}(x,y,z)=\alpha\int\int\frac{z\, dx'\, dy'}{((x-x')^{2}+(y-y')^{2}+z{}^{2})^{3/2}}$$ with the limits $$-\frac{a}{2}\leq ...
1
vote
3answers
69 views

Evaluating an integral in physics question

$$U_{C} = \frac{1}{C} \int\!\frac{\cos(100\pi t + \pi/4)}{10}\,dt$$ Find $U_{C}$, the answer is $U_{C}=\left(3.2\times 10^{-4}\right)/C\times \cos(100\pi t - \pi/4)$. Can someone show to to get ...
0
votes
2answers
71 views

Work done by a force Field

Homework for Calc III includes a problem about computing the work done by a force field (defined by a specific vector equation) on a moving particle. I was attempting to compute this using the ...
2
votes
1answer
144 views

Trouble understanding a common vector calculus example

I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
1
vote
1answer
64 views

Minimum Ejection Velocity

PROBLEM: Calculate the minimum ejection velocity with which a shell must be fired to strike a target 1000ft high and directly overhead. QUESTIONS: I use integration to work back from 32ft/sec and ...
1
vote
1answer
177 views

Moment of inertia of a circle

A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$ Thank you
1
vote
1answer
258 views

Electrostatic Potential Energy

How is the boxed step , physically as well as mathematically justified and correct ? Source:Wiki http://en.wikipedia.org/wiki/Electric_potential_energy As work done = $- \Delta U $. for Conservative ...
8
votes
1answer
302 views

Integral - Problem

Hi I am stuck with an integral problem trying to show $$ -\int \frac{d^{3}p}{(2\pi)^3}p\frac{\partial f(p,t)}{\partial p}=3F$$ where $F=\frac{1}{(2\pi)^3}\int d^{3}pf(p,t)$ I have read in many books ...
1
vote
0answers
44 views

Taylor expansion of an integral in spherical co-ordinates

I've some difficulty deriving this equation from jackson electrodynamics (The equation after 1.30) $\nabla^2 \Phi_a\left({\textbf{x}}\right)=-\frac{1}{\epsilon_0}\int_{0}^{R} ...
1
vote
1answer
1k views

Using a Fourier Transform and Parseval's Theorem to Solve an Equation

I'm sorry to bother you, but I've been studying for a test, and I kinda got stuck in this question. Let me place the question then tell you what I've done so far. I have to find the Fourier Transform ...
5
votes
1answer
172 views

Is electrostatic energy positive definite?

This is a question coming from physics, but its nature is purely mathematical. Given some continuous distribution of charge $\rho$ (take it compactly supported, or "nice enough" depending on the ...