Volume Definition and 1000 Threads

Integration in the order of dy then dx: $$\int_{-1}^{1} \int_{0}^{1-x^2} x^2 \sqrt{1-y} ~dy~dx$$ $$=\int_{-1}^{1} -x^2 \left[\frac{2}{3} (1-y)^{\frac{3}{2}}\right]_{0}^{1-x^2}dx$$ $$=\int_{-1}^{1}\left(-\frac{2}{3}x^5 + \frac{2}{3} x^2\right)dx$$ $$=\left. -\frac{1}{9} x^6 + \frac{2}{9}...

I got the two relations for spherical and rectangular coordinates. In rectangular...

Is the number of Higgs "particles" per unit volume constant (or Higgs field value), even in expanding space?

Hello, how do we calculate the volume of a hollow cylinder but only calculating the wall excluding the hollow center?

Until very recently I thought that according to general relativity masses curve spacetime as a whole, i.e. the complete abstract 4-dimensional spacetime. It came as a complete surprise to me to learn that you can take away the time-component and are left with a curved 3-dimensional space, which...

I'm a little thrown off with material I'm going through right now. I already covered the whole "area under the curve" and using that to determine the volume of a given equation, but I'm confused now as to why calculating the surface area has a different method with ds? For example, say there...

I am trying to calculate the size of coolant tank required to feed multiple grinding stations. I have a total of 12 taps all with flow rates of 20L per minute. All taps would operate at the same time. The tank would be open topped and 500mm tall. The tank also requires a dirty side and a...

If you had pressure sensors (red) inside closed containers (black). The one on the left having more air volume than the one on the right. They are inside a bucket (green) that is filled with water. The water will rise higher in the left container. Will the pressure sensor readings be identical...

Hi, 1st and general case: 2 H2 and 1 O2 take up 1 liter of volume at, for example, 20°C, before combustion. After combustion and cooling down to 20°C, how much volume do the exhaust gases take up? Since the stoichiometric relationsship is 2 mols of H2 for every mole of O2, so those three moles...

PV=nRT P*1L = 6 moles * 0.0821*298 (I added up all the moles and solved for pressure) P =146.79 atm 146.79 atm * V = 2 moles Ar * 0.0821* 298 (I plugged in the moles for argon and solved for volume) V= 0.333 L Answer key says the answer is 1 Liter. Where did I go wrong?

Hi. Since I've been studying fluid mechanics, I've been asking a lot of questions Thank you all for your help! I suddenly have a problem that I want to think about while studying today. I would like to know the change of air volume according to the pipe shape as shown in the picture below...

Hello I need your help studying hydrodynamics. I have a question about the volume of air. Assuming incompressibility, non-viscosity, there is a coefficient of tube friction, and it moves in laminar flow... The Oulet stage Fan is installed, so air moves from top to bottom. The cross-sectional...

Like the title and the summary suggest, I can derive the volume ##V=2\,\pi^{2}\,r^{2}\,R## for a ring torus - a doughnut-style toroid (one such that the major radius ##R## > the minor radius ##r##, and it therefore has a hole at the center) that is of circular cross-section. But I want to be...

Hello everyone, I was trying to find volume of a sphere by doing some calculus, the area of a circle is ##{\pi}r^2## So I thought I would calculate the volume of one hemisphere and then multiply by two, but I got a different result than the standard formula, the standard formula is ##\frac 4 3...

When a photon interacts with matter and diffracts, the photon seems to interact with the area of its wave front, determined by an aperture even if it is only one photon. But how far does it interact in depth? Does it always "feel" the whole crystal? Can we assume the photon is reflected by the...

Is it correct that: density = [ 0.09 * (density of NaCl) ] + [0.91 * (density of water) ] volume = [ (volume of water) + ( { (volume of NaCl) / 48} * 9) ] Thank you so much for any advice.

Butane 0.75mol of C4H10 0.75 x C4 = 3mol of CO2 0.75 x H5 = 3.75mol of H2O 0.75 x 6.5O2 = 4.875mol of O2 Propane 0.10mol of C3H8 0.10 x C3 = 0.3mol of CO2 0.10 x H4 = 0.4mol of H2O 0.10 x 5O2 = 0.5mol of O2 Butene 0.15mol of C4H8 0.15 x C4 = 0.6mol of CO2 0.15 x H4 = 0.6mol of H2O 0.15 x...

Hey! Im currently writing a lab on how an increase in the volume of an object will result in an increase of buoyancy force acting on an object. We fully immerse different amounts of clay playing blocks into water (using a string) on top of a scale, and calculate the buoyancy force. The reasoning...

Also, $$V=\dfrac{1}{4} \int_4^0 \left[\int_{2-0.5y}^2 (4-2x-y) dx\right] dy$$ $$V=\dfrac{1}{4} \int_4^0\left [4x-x^2-xy]^2_{2-0.5y} \right] dy$$ $$V=\dfrac{1}{4} \int_4^0 \left [(4-2y)-(4(2-0.5y)-(2-0.5y)^2-(2-0.5y)y] \right] dy$$ $$V=\dfrac{1}{4} \int_4^0 \left [(4-2y)-(2-0.5y)^2 \right]...

Hi all, I've made errors in my calculations and need help. I'm trying to calculate the volume of a layer above sea level to the height of Flight Level 300 i.e. 30,000 feet. Next, I want to subtract the volume of land within that layer. Here are my calculations. Purely for the sake of my...

My attempt: The shape of the hyperboloid would be like this: If the hyperbolod is cut by plane z = d, the intersection would be a ellipse. Projecting the intersection to xy - plane, I think I get: $$-2\leq x \leq 2$$ $$-b\sqrt{1-\frac{x^2}{a^2}} \leq y \leq b\sqrt{1-\frac{x^2}{a^2}}$$ So the...

Good evening, I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context. I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this...

I am sure that the radius of the two spheres changes equally. But in answer of the question it said that their change in volume is the same. Is it correct?! Link of website: https://www.toppr.com/ask/question/two-spheres-of-same-size-are-made-of-the-same-metal-but-one-is-hollow/ I think it is...

m * g = mAl * g V * ρ * g = VAl * ρAl * g V * ρ * g = V * ρAl * g ρ = ρAl this does not work at all, because the upper ball must have a density smaller than that of seawater 1200kg/m3 or not?

I am given this expression which represents an object in 3D and the goal is to determine its volume using multiple integrals. I started by drawing what I think is the object as well as two "slices" of that object on different planes (z=2 and z=1) I have tried using cartesian, cylindrical and...

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...

An insulated container (constant volume, adiabatic) contains an Ideal gas with pressure P1 and temperature T1. We open the container's hatch for a few seconds and let some particles escape from the container, then we close the hatch again. We know container's pressure has reduced by exiting...

Hi everyone, I am struggling to understand whether the results of FMESH tallies are already divided by the cell volume or not. I'd actually expect so considering: 1. the comparison with an F4 tally in the same cell where results are comparable only if I assume that the mesh tallies results are...

I have a cube with a volume of 1000m3 at an initial temp of 290K. The bottom side (10m by 10m) is open to the ambient air. I put this cube into a huge fridge and cool the whole volume by 5K. I close the open side by placing a cover on it. This cube has now got a volume of air at a temperature of...

If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above the surface of water. What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant). It's easy problem but I can't get right...

Using the adiabatic process formula, I've calculated the change in volume for a volume of gas with an initial pressure of 10 psig expanding to 0 psig. The initial volume is 100 cubic inches and the expanded volume is 144.9. This is a difference of 44.9. The total work done ends up being about...

I consider a disc of thickness ## R d\theta ## as shown in the figure. Then, $$ dV = \pi R^2 sin^2 \theta R d\theta $$ ( Area of the disc * its thickness) Hence, $$ V = \int^{\pi}_{0} \pi R^2 sin^2 \theta R d\theta $$ $$ V = \frac 1 {2} {\pi}^2 R^3 $$ ....(1) While $$ V =...

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So I'm reading through Cengel's thermodynamics textbook, and came across this solved example: Firstly, pressure in this context I'm assuming is vapour pressure? Since we're dealing with pure substances in this chapter. But what's confusing me is, here's the diagram I have: They've not...

I have a volume of 0.04 cubic centimeters. I need to convert into something I can work with in my shop. Tolerance is not important. To get dimensions of a cube with this volume, I calculated the cube-root which came out to 0.341. So, my cube would be (rounded) 3mm on a side, correct? Next...

has been resolved Please help me to understand which answer is correct. I use the integral method to find the elbow volume as follows： But my book say：

In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following. $$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$ $$ \bf z = \bf r^{''} - \bf r^{'} $$ $$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z +...

I have been away from thermodynamics for a while, and there is a young mechanical engineering student who asked my help in solving the question. For the life of me I am unable to solve this problem and I am at my wit's end. I believe I can solve the problem if some pointers are provided. Also...

So the question goes like this: find change in internal energy in process 1->2 using diagram. And offered solutions a)-400J b)400J c)600J d)800J. First I found T1 and T2 using (P*V)/T=R and got T1=24K and T2=72K. Then I found n(number of moles) using PV=nRT and got n1=1mol, n2=1mol. Then I...

I was reading an article by Edward Harrison, which tackles the problems of conservation of energy at cosmological scales. At some part (point 2.4) he cites several article, including one by Rees and Gott, which he says indicates that the internal energy of a comoving volume (e.g. a cosmic...

Part (a) was simple, after applying $$Q=\int_{\mathbb{R}^3}^{}\rho \, d^3\mathbf{r}$$ I found that the total charge of the configuration was zero. Part (b) is where the difficulties arise for me. I applied $$V(\mathbf{r})=\frac{1}{4\pi \epsilon _0}\int_{\Gamma }^{}\frac{\rho...

I have a storage tote that has a larger top than bottom. How do I figure out its volume? Is this like a 3D trapezoid? Can I measure the volume of the rectangle assuming the top is the same as the bottom, then as if the bottom were the same as the top, then just subtract the two?

I can’t quite work my head around this question, I am having a difficult time analyzing the question, I can’t seem to make out what the initial and final conditions are would appreciate all the help I could get cheers

At a single moment of coordinate time ##t##, would a distant observer perceive a warp bubble's interior volume as blown up, or would it seem compressed? Looking in the catalogue of spacetimes at the static local tetrad of the Alcubierre metric, the ##e^x_{(x)}## leads me to think that a static...

I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...

Using the equation above, I plugged in 5.5 inches for the radiu and 0.5 inches for the value of dr and then solved for the estimate of the change in volume, dV. However, the solution instead uses a value of 6 inches for the radius receiving a different estimate for the problem than I did. Is my...

I'm getting the wrong results when using an old, undocumented code and just realized there's a number lurking in it that I can't account for. It's: 0.023901488 and it is multiplied with molar volume and pressure. I have searched for a couple of hours but just can't figure out what the units...

Problem statement : I draw the problem statement above. I hope I am correct in inferring that the bowl is hemispherical. Attempt : I could not attempt to the solve the problem. We are given that the rate of change (decrease) in volume is proportional to the surface area ...

In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents) \begin{align} \vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3}...

So, let's assumed ##v_i = (v_i \pm \Delta v_i)##. We can say that, for minimum values, $$\Delta X_i \Delta v_i = \hbar / (2m)$$. $$\Delta X_i = \frac{\hbar}{2 m \Delta v_i} \implies V_{min} = \Delta X \Delta Y \Delta Z = \frac{\hbar}{2 \Delta v_x m } \frac{\hbar}{2 \Delta v_y m} \frac{\hbar}{2...