# Tagged Questions

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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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### 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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### Moment of inertia of a n-dimensional sphere

Is it well known that the moment of inertia of a sphere can be calculated starting from its density $\rho=\frac{m}{V}$ where $V$, the volume of the solid is: $V=\frac{4}{3}\pi r^3$. Calling it $I_3$, ...
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### Zero velocity field inside an ellipse

I'm investigating the velocity field induced by a continuous distribution of 2D vortex points distributed along an ellipse $\{a\cos\theta,b\sin\theta\}$. I'm interested in the field inside the ...
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### Is $f''(f)df=f'(x)df'$ correct?

This question is about kinematics and $a , v, x$ stand respectively for acceleration, velocity, position. Supposing we have an expression of $a$ in function of $x$, we have the following theorem: ...
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### Help with integral for electric potential

I need help evaluating the following integral $$\frac{ \sqrt 2 \sigma}{2 \epsilon_0} \int_0^R \frac{r \,dr}{ \sqrt{(z- \frac{ rh}{R} ) ^2 + r^2} }$$ This integral pertains to the Electric potential ...
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### Surface area , center of mass, and moment of inertia of paraboloid

I have done a bunch of work and simply wish to check that it makes sense. I have a hollow parabola of height b and base radius b ($z = \frac{x^2 + y^2}{b}$ bounded by z = b) 1) surface area of ...