Questions tagged [volume]

For questions related to volume, the amount of space that a substance or object occupies.

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19 views

volume of a block with known are of a square inside

The task is to figure out the volume of a block ABCDEFGH . AB(EH ...) is x and BC(also FG...)is x + 23. The third side is unknown(AE, BF ...) The area of the rectangle BCEH is 4225. sketch
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1answer
17 views

What is the answer to this question?

A cylinder with closed ends has a total surface area 𝑆; the radius of the base is π‘Ž and the height is π‘˜π‘Ž. Find an expression for π‘Ž in terms of π‘˜ and 𝑆. The expression I get cancels down to $√a^...
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32 views

What is wrong with my answer to the question below?

A cylinder of radius $𝑅$ contains some liquid; a solid metal cuboid with sides $π‘Ž$, $π‘Ž$ and $2π‘Ž$ is totally immersed in the liquid. Find an expression for the decrease in the height of the liquid ...
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1answer
39 views

Volume using double integrals

Calculate the volume of the solid bounded by the following surfaces: $y=x^2, y=1, x+y+z=4, z=0$. How does on set up the integral?
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3answers
46 views

Find the volume between the surface $x^2+y^2+z=1$ and $ z=x^2+(y-1)^2$

I'm trying to find the volume between the surface $x^2+y^2+z=1$ and $ z=x^2+(y-1)^2$ but nothing works for me. I made the plot and it looks like this: How could you start? Any recommendation?
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2answers
27 views

How to find a volume of an object enclosed with planes without any projection?

How to find a volume of an object enclosed with planes: $$x^2+z^2=4,$$ $$x+y=2,$$ $$x+y=-2,$$ $$x-y=2,$$ $$x-y=-2$$ without any projection? When I project this object I know it is a truncated ...
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2answers
53 views

Finding a volume of a region defined by |x-y+z|+|y-z+x|+|z-x+y|=1

Find the volume of the region definded by |x-y+z|+|y-z+x|+|z-x+y|=1. I'm having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful. Thanks in ...
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2answers
59 views

Density definition implies total mass integral

Reading back through an old text on E&M today, I realized that while it makes intuitive sense to me that upon defining density as $$\rho(x,y,z) = \frac{dm}{dV}$$ we get that $$M = \int_V \rho dV$$ ...
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19 views

Volume of the solid obtained by revolving

To see the question please click here I found the function is second root of $(x-1)$. The question I can't find the answer is that: Find the volume of the solid obtained by revolving the region $...
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1answer
37 views

Calculus word problem involving volume of solid

I'm trying to solve this calculus problem: A tower is constructed with a square base and square horizontal cross-sections. Viewed from any direction perpendicular to a side, the tower has base $y = ...
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1answer
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Finding The Volume Of a Shape That Is Given By the Formula $3x^2 + 2y^2 + z^2 \leq 6$

how can I find the volume of the shape that is given by the formula $3x^2 + 2y^2 + z^2 \leq 6$ ? Thanks!
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1answer
28 views

Volume by Rotation Using Integration

How to find the volume if the shown area is rotated around the $y$-axis? The area will be bounded by $𝑦=π‘₯^2+1$, $y=2x$ and $x=0$.
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26 views

Lipschitz maps cannot increase the volume of a Borel set by a factor greater than $k^n$.

Let $(M,\,g,\,d,\,vol_g)$ be a Riemannian manifold with metric $g$, geodesic distance $d$ and volume form measure $vol_g = \sqrt{\det(g_{ij})}\cdot m$ ($m$ = Lebesgue measure) and a Lipschitz map $f:K\...
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1answer
25 views

Find the volume of a specific region of a solid of revolution about the y-axis

I have a given polynomial, f(x)=c1*x^7+c2*x^6+c3*x^5+c4*x^4+c5*x^3+c6*x^2+c7*x+c8 rotated about the y axis, which results in a nice surface: What I would like to be able to do is evaluate the volume ...
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2answers
73 views

Calculate the volume of the solid determined by $S_1$ and $S_2$

I want to calculate the volume of the solid determined by this tho surfaces: $$S_1=\{(x,y,z)\in\mathbb{R}:x^2+y^2+z^2=R^2\}$$ $$S_2=\{(x,y,z)\in\mathbb{R}:x^2+y^2=Rx\}$$ The solid is the intersection ...
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Finding Volume of a Torus (Spherical Coordinates)

Question: Find volume of region bounded by torus where equation of torus in spherical coordinates $(r, \theta, \phi)$ is $r=2 \sin \phi$ for $\phi \in [0, \pi]$ and $\theta \in [0, 2\pi]$. Attempt: ...
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volume of cone,cylinder, hemisphere

volume of cone,cylinder,hemisphere So i find volume of hemisphere:2/3*Ο€r^3 volume of cylinder is Ο€r^2*h volume of cone is 1/3*Ο€*r^2*h now how to proceed to solve, calculate h(height),and use the ...
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1answer
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Curse of dimensionality $2^d +1$ hyperspheres inside a hypercube

Consider a $d-$dimensional hypercube $Q$ of side length $l ∈ R, i. e. |x_i βˆ’ y_i | ≀ l$ for all $x, y ∈ Q$ and all $i ∈ [d]$. Note that $Q$ has $2^d$ corners. We fill $Q$ with ($L_2-)$hyperballs the ...
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2answers
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Can you see clearly _why_ $x$ and $h$ are linearly related in a triangle?

Consider a triangle with $x \in (0,6)$ and $h \in (0, 8)$: Question: Often times in questions related to volumes I have leveraged the property that $x$ and $h$ are linearly dependent to get the ...
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33 views

Find the volume of the solid obtained by rotating

Let R be the region in the first quadrant bounded by the curves $y = f(x) = 2x+ 1$ and $y = g(x) = 2x^2 βˆ’8x + 9$ Find the volume of the solid obtained by rotating the region R about y-axis using ...
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20 views

A new space satellite has a smooth unbroken skin made up of portions of 2 circular cylinders of equal diameters $D$ whose axes meet at right angles.

A new space satellite has a smooth unbroken skin made up of portions of 2 circular cylinders of equal diameters $D$ whose axes meet at right angles. It is proposed to ship the satellite to Cape ...
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1answer
23 views

Octant issue with Sphere, and Elliptic Paraboloid

This is the graphical setup for my problem Math3d. I given a sphere in cylindrical coordinates as the following \begin{equation}r^2+z^2=20 \end{equation} Which is the equation of a sphere of radius $\...
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29 views

General idea for volume of $n$-ball via integrals

If I have some $\mathbf{x}\in\mathbb{R}^N$, and $\mathbf{x}^{\intercal}\mathbf{x}\leq C = \mathcal{C}$. Is the following integral by definition the volume of an $n$-ball? \begin{align} \int_{\...
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1answer
28 views

Finding volume of solid in one quadrant - divide total volume by 4? 8? 2?

I want to find the volume of the solid produced by revolving the region enclosed by $y=4x$ and $y=x^3$ in the first quadrant. The wording about the first quadrant confuses me but here's my work so far:...
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19 views

The sum of the volumes of two non-overlapping sets

The problem is to prove that $\text{Vol}(E_1)+\text{Vol}(E_2)=\text{Vol}(E_1\cup E_2)$ for any non-overlapping Jordan regions $E_1,E_2\subseteq \mathbb R^n$. There is one inequality in my book's proof ...
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18 views

Finding the Volume of a Torus by rotating a circle with equation $x^2 + (y-R)^2 = r^2$ a different way

I was trying to find the Volume of a Torus and after sketching it out in Desmos, I thought I found a solution but it doesn't work (or at least when I evaluated it it didn't work). $V = 2Ο€\int_{-r}^r(...
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2answers
14 views

Volume of Solids of Revolution with Hyperbola

The area bounded above by the line $y = 3$, below by the line $y = 0$, on the left by the y-axis and on the right by an arc of the hyperbola $9x^2 - 16y^2 = 144$ is rotated around the x-axis. Find the ...
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27 views

Double/Volume integral in parametric form

Find the volume of the tetrahedron with vertices at $(0,0,0),(a,0,0),(0,b,0),(0,0,c)$ The most straight forward approach would be evaluating the following integral in Cartesian coordinates system. $$...
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1answer
79 views

Integral of product of 3D Gaussians

I am reading a paper, Keep it SMPL: Automatic Estimation of 3D Human Pose and Shape from a Single Image, CVPR 2016, that models the human parts with capsules for estimating the interpenetration ...
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Finding Volume of a solid via integration

I'm new to integration and I have this question I have to work out. Please let me know if I went wrong. Thank you. I have to find the volume of a shape by the bounds below, about the y-axis: $$ y = ...
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1answer
33 views

Triple integral and mass

An object occupies the solid region bounded by the upper nappe of the cone $$z^2=9x^2+y^2$$ and the plane $$z=9$$ Find the total mass of the object if the mass density at (x,y,z) is equal to the ...
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1answer
27 views

find the volume using the method of disks or washers via an integral

The volume of the solid obtained by rotating the region enclosed by $y=\frac{1}{x^2} , y=0, x=3, x=8$ about the line $y=-1$. How do I find the volume? I need help. I tried $\pi\int_8^3(\frac{...
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36 views

3-dimensional surface volumn of a 4-dimensional hyperellipsoid

The circumference of a 2-dimensional ellipse and the surface area of a 3-dimensional ellipsoid can be given in close-forms using the Legendre’s elliptic integrals as $$C_2=4aE(\sqrt{1-b^2/a^2})$$ $$...
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2answers
59 views

Calculate volume of $(x^2+y^2+z^2-az)^2=R^2[x^2+y^2+(z-a)^2]$ [closed]

Give a surface $(x^2+y^2+z^2-az)^2=R^2[x^2+y^2+(z-a)^2]$, R,a is constant, $0 < a < R$, how to calculate the volume that is contained in this surface? I know the answer is $V=\frac{4\pi}{3}R(R^...
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How do I prove Guldin Pappus theorems for area and volume using mathematical analysis? [closed]

I don't know how I can prove these 2 theorems. Can anyone help me, please?
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2answers
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I need help solving this integration problem. [closed]

The base at a solid is the region between the curves $f(x)=x^2-1$ and $q(x)=1-x^2$ , and its cross-section perpendicular to the x-axis are equlateral triangles.Find the volume of the solid.(Plot the ...
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3answers
52 views

Volume and surface area of $3/4$ of a sphere

Take for example a 3-D sphere cut horizontally into quarters: How would I identify the volume and surface area of top $3$ horizontal cuts? Would it just be $\frac34\cdot$volume of complete sphere ...
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1answer
11 views

If the region $D$ is revolved about the $z$-axis in $ℝ^3$, then the volume of the resulting solid is

Consider the region $D$ in the $yz$ plane bounded by the line $y=\frac{1}{2}$ and the curve $y^2+z^2=1$, where $y\geq 0$. If the region $D$ is revolved about the $z$-axis in $ℝ^3$, then the volume of ...
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22 views

which has a greater volume - unit sphere or a tetrahedron with given vertices?

I know the volume of a unit sphere is $\frac{4\pi}{3}$ (using integration) but I don't really know how to evaluate the volume of a tetrahedron with vertices: $$(0,1,1), (0,-1,1), (-\pi,0,-1), (\pi,0,-...
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2answers
29 views

Calculation of the volume of a chamber in a rotary vane pump

Question proper: What is an expression for the area shown in grey in the diagram below? I wish to ascertain the volume of a chamber in a RVP (Rotary Vane Pump). (Volume is proportional to area in ...
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1answer
37 views

Volume enclosed by $(x^2+y^2+z^2)^2 = a^2(x^2+y^2-z^2)$

My math problem is the find the volume enclosed by the surface $$(x^2+y^2+z^2)^2 = a^2(x^2+y^2-z^2)$$ I used spherical coordinate substitution, $$\left\{\begin{matrix}x=\rho\cos{\phi}\cos{\theta}\\ y=...
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2answers
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What's the way to know how many 1mm$^3$ falls into 9.0 $\times$ 10$^9$ L? [closed]

How to easily calculate how many 1mm$^3$ falls into 9.0 $\times$ 10$^9$ L?
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1answer
23 views

Evaluate the Double Integral at the Following Region

Problem Statement Evaluate $\iint_R xdA$, where R is the region bounded by $x=\ln(y)$, $x=0$,$y=e$. My Work: $$\int_0^e\int_0^{\ln(y)}x\ dx\ dy$$ or $$\int_0^1 \int_{e^x}^e x\ dy\ dx$$ My Problem ...
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What is meant by rate of change with respect to volume?

In physics we often come across $$\rho=\dfrac{dq}{dV}$$ Does it mean: $(i)$ $\displaystyle \lim_{\Delta V \to 0} \dfrac{\Delta q}{\Delta V}$ OR $(ii)$ $\dfrac{\partial}{\partial z} \left( \dfrac{\...
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2answers
49 views

Using Mass and Density of a Sphere

Question: A sphere of radius $R$ has total mass $M$ and density function given by $ρ = kr$, where $r$ is the distance a point lies from the centre of the sphere. Give an expression for the constant $k$...
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32 views

Calculating areas vs volumes in multivariable calculus

The formula for the area is: $$\iint_RdA$$ The formula for the volume is: $$\iint _R (f(x,y)-g(x,y))\,dA$$ In other words, the volume is surface base's area * height, or the base repeated over and ...
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1answer
21 views

Question about Polar Coordinates and Constant Values

My questions pertains to why the following limits of integration would be the following given the following scenario:\begin{equation}\int_0^a \int_0^\sqrt{a^2-x^2} \frac{dy\ dx}{(1+x^2+y^2)^\frac{3}2} ...
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11 views

Is there a technique to identify the r limits, and function which is being integrated?

I am given the following example of which I have the solutions the already. I just wonder how the regions which are in this particular format: \begin{equation}\iint_Rf(r,\theta)r\ dr \ d\theta\end{...
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1answer
25 views

Where does the formula for the volume of a hemisphere come from?

Where does the formula for the volume of a hemisphere come from, I have been using this The formula
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1answer
41 views

How to calculate the volume between a paraboloid and a parabolic cylinder

So I have just started studying double integrals and the book I have bought hasn’t many solved examples. An exercise asks to show that the volume of the solid created by the intersection of the ...

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