0
votes
1answer
71 views

Finding total work by integration

The following tank is completely filled with water. Find the total amount of work done in pumping water out of the outlet. Note that the density of water is 1000 kg/m$^3$ I feel like I am ...
1
vote
1answer
147 views

Work on Springs using Hooke's Law

I'm currently stuck on parts c and d of this problem. The problem says Suppose a force of 20 N is required to stretch and hold a spring 0.4 m from its equilibrium position (0). I found k constant to ...
0
votes
0answers
25 views

Solving the integral to calculate the electrostatic force between two tubes

Let's say we have two tubes with charges $q_1$, $q_2$, radii $b_1$, $b_2$ and lengths $l_1$ and $l_2$. They are placed along the surface of each other like in this figure: To calculate the ...
0
votes
0answers
34 views

Testing Divergence Theorem using Spherical Coordinates

I'm trying to verify the divergence theorem using spherical coordinates for the vector field $\vec{F}=r^2cos^2\theta(cos\theta\hat r-sin\theta\hat\theta)$ through the top half of the unit sphere. ...
1
vote
0answers
50 views

How can I calculate the force that is applied on a tube by an another tube?

Let's say there is two tubes (cylinders with no tops or bottoms) with charges $q_1$ and $q_2$, radii $b_1$ and $b_2$, lengths $\ell_1$ and $\ell_2$. These tubes are located along the axis of each ...
2
votes
2answers
51 views

Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...
0
votes
5answers
102 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
1
vote
1answer
24 views

Setting Up an Integral (depends on displacement vector from a long line)

The following is a physics problem, but I don't actually care about the physics right now. I just want to set up the integral, which I'm having a difficult time doing (the problem is from the book ...
1
vote
2answers
48 views

Constants for anti-derivatives

Hey StackExchange I'm diving into integral calculus for the first time and I have a few questions about this problem. A steel ball bearing at rest is accelerated in a magnetic field in a line with ...
-3
votes
1answer
40 views

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

I want to find the integral of: $$\int_R^\infty \frac{GMm}{r^2}\,dr$$
1
vote
1answer
42 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
43 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
29 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
63 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
44 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
54 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
68 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
42 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
44 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
78 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
49 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 ...
5
votes
2answers
128 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
50 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
45 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
94 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
51 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
103 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
109 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
53 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
92 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
31 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
131 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
399 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
188 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
66 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
38 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
3k 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
727 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
92 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
197 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
128 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
365 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
334 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
489 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
71 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 ...