# Tagged Questions

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### 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 $\Delta x$ ...
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### 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 ...
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### Taking cross product out of integral

Im trying to show that the integral over a closed loop of a crossproduct stays the same if I choose a different origin with $\overrightarrow{r}=\overrightarrow{r}\prime+\overrightarrow{r_0}$ and ...
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### 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 $$\int_S x_1 \, dx_1 \, dx_2=0$$ $$\int_S x_2 \, dx_1 \, dx_2=0$$ ...
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### 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 ...
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### 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 ...
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### 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. ...
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### Hankel trasformation of acoustic wave equation

We consider a simplified version of acoustic wave 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 ...
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### 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 ...
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### Help with explaining momentum relation $d^3 p = 4\pi p^2 \,dp$ [closed]

I need some help deriving the following relation! $$d^3 p = 4\pi p^2 \,dp$$ Thanks!
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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 ...