1
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0answers
43 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
92 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 ...
1
vote
0answers
16 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=\int_0^l\int_0^l ...
0
votes
1answer
36 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
63 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
22 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
58 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
54 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
35 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
79 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
126 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
209 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
138 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
101 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
43 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
32 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
761 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
45 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
1k 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
442 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
79 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
161 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
114 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
263 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
307 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
332 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
67 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
67 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
138 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
57 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
160 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
249 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
298 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
985 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
167 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 ...
2
votes
1answer
41 views

When does $\int_a^b \frac{dx}{1-x^2} \neq Ln(1 + x) - Ln(1 - x)?$

I have an integral Over a particular bound. $$\int_{V_o}^{V_f} \frac{dv}{1 - \frac{v^2}{v_t^2}}$$ Can this integral Be that of an Inverse Hyperbolic Tangent, even though it looks like it can be ...
6
votes
0answers
187 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
9
votes
1answer
163 views

Help computing an integral for Green's function of a discrete Laplacian on a square lattice

I need to calculate the following integral: $$ \int_0^1 \int_0^1 \frac{1-\cos(2 \pi k_1 x) \cos(2 \pi k_2 y)}{4 \sin(\pi k_1)^2 + 4 \sin( \pi k_2)^2} dk_1 dk_2 $$ I have tried to use some contour ...
3
votes
0answers
115 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
4
votes
0answers
107 views

Gaussian Integral with non-polynomial exponent

I am currently trying to evaluate this Integral: $$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$ Where ...
4
votes
1answer
132 views

Existence Energy of Wave Equation

I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential ...
2
votes
2answers
72 views

Hint for integral

Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec ...
3
votes
3answers
987 views

Cat Dog problem using integration

Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation ...
5
votes
4answers
484 views

Numerical approximation of a complex integral with a nested exponential

I've been working on a maths problem as part of my Physics PhD; but have been stumped by the following integral. All I need to know is a numeric approximation to the integral (along with an estimation ...
1
vote
1answer
19 views

Simplification of Equation Involving Second Partials

I was reading this article and I'm trying to follow this author's proof. The author jumps from $$\psi_1(x)\frac{\partial^2\psi_2(x)}{\partial x^2}-\psi_2(x)\frac{\partial^2\psi_1(x)}{\partial ...
2
votes
2answers
3k views

A trough is 3 feet long and 1 foot high. The trough is full of water…

A trough is 3 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of $x^2$ from -1 to 1 . The trough is full of water. Find the amount of ...
1
vote
1answer
127 views

What does it mean mathematically to set some of the integration constants in the general solution to a linear differential equation, equal to zero?

I'm trying to calculate the position of a particle in a quadrapole magnet depending on the entry position $x_0$ and the combined (constant) physical parameters $k$. Given an equation $$x(t) ...