Questions tagged [volume]
For questions related to volume, the amount of space that a substance or object occupies.
3,060
questions
0
votes
0
answers
39
views
Find the volume of the given solid
Find the volume of the solid
$$\{(x,y,z) : x ≥0,y ≥0,z ≥0,z ≤2 −e^y, x + y + z ≤10\}.$$
Not sure if my workings for this is correct? Any help would be appreciated, thanks!
workings
1
vote
1
answer
56
views
Calculating Failure Angle of Silo Wedge
I am working on an optimization problem, specifically calculating the lateral earth pressures on the inside of a silo wall. I have been trying to solve for the failure angle of a silo wedge based on ...
- 11
1
vote
1
answer
25
views
An improper integral and a related volume calculation
I am trying to solve Whittaker & Watson's Chap 6, Misc Example 6.13. I understand the formula to be
$$
\int_0^\infty \left( \frac {\sin t} t \right)^n \,dt = \frac{\pi}{2^n (n-1)!} \sum_{k=0}^{\...
- 95
2
votes
1
answer
43
views
Surface area - even dimensions
Consider the surface:
$$ (\log x_1)^2+(\log x_2)^2+\cdot\cdot\cdot +(\log x_n)^2=R $$
For $R=1$ and in even dimensions $n=2,4,6, \cdot\cdot\cdot$ we have the volumes:
$$V=\bigg( \frac{\pi^2 I_1(\sqrt{...
- 196
0
votes
1
answer
31
views
Largest hyperrectangle contained in an ellipsoid
I am looking for a hyperrectangle with maximum volume which is contained in an ellipsoid.
My Ellipsoid is defined as
$\mathcal{E}=\{x\in\mathbb{R}^n\mid x^T P x \leq c \}$, where $P>0$.
For smaller ...
- 63
0
votes
2
answers
79
views
Find the volume of $\mathbb S^n$ (the surface area of $(n+1)$-dimensional unit ball) by using differential geometry
I am reading the first 16 chapters of Lee's Introduction to Smooth Manifolds and trying to find the volume of $\mathbb S^n$.
I know the Riemannian volume form of $\mathbb S^n$ is
$$\omega=\sum_{i=1}^{...
55
votes
3
answers
5k
views
Why isn't the derivative of the volume of the cone its surface area?
The derivative of the volume of a sphere, with respect to its radius, gives its surface area. I understand that this is because given an infinitesimal increase in radius, the change in volume will ...
- 884
0
votes
1
answer
31
views
Calculate the volume of the body formed by the rotation of the figure
I need to calculate the volume of a body formed by rotating a figure bounded by curves $y=0$, $x=\sqrt{1-y^2}$ and $y=\sqrt{\frac32}x$ around the $Ox$ axis.
The graph of the function and the required ...
0
votes
1
answer
52
views
Instant rate of growth of a spherical balloon
I am having trouble with a problem involving a spherical balloon, and I was hoping someone could help me out. The problem is as follows:
"Air is blown into a spherical balloon so that its volume ...
- 111
3
votes
1
answer
56
views
Volume Under a Ceiling
I have run into an interesting volume problem that I can't seem to figure out. I was recently in a school gym and noticed the interesting design of the ceiling and thought back to Calculus III and ...
- 31
0
votes
1
answer
52
views
How to get the Volume of an irregular 3d shape [closed]
I am calculating the shear and moment for a combined foundation with bi-axial moments on each columns.
I got the pressure at 4 corners under the foundation.
in order to be able to calculate the sheer ...
- 103
0
votes
1
answer
19
views
Find the volume of a function that revolves by pi radians.
I know that the volume of a function which revolves around an axis by a full turn is the following:
$$V=\pi \int_a^b f^2(x)dx$$
But I'm confused as to what to do when the function is asked to be ...
0
votes
0
answers
27
views
Volume form in Finsler Geometry
Why this volume form does not hold for the Busemann-Hausdorff volume form.
- 13
0
votes
1
answer
31
views
Find the volume of a solid of revolution by revolving the region bounded by the $x$-axis and the graph of $y=x\sqrt{2-x}$ about the line $y=−5$.
From what I understand, I am supposed to convert everything to terms of $y$, but I'm not sure how to solve for $x$ in $y=x\sqrt{2-x}$.
So far I have the setup of: $$2 \pi \int_0^{\frac{4 \sqrt{6}}{9}}...
- 25
0
votes
1
answer
16
views
$n$-dimensional spheres and gaussian scalar fields
I'm currently reviewing some problems in Statistical Mechanics and I have come across a question that I'm struggling to resolve. Specifically, in certain parts of the study of ideal gases, the concept ...
- 153
1
vote
1
answer
55
views
Finding the volume between sphere and hyperboloid
I am trying to find the volume between the sphere: $x^2+y^2+z^2=9$ and the hyperboloid $x^2+y^2-z^2=1$. I set the integral as: $$\int_{0}^{2\pi}\int_{-3}^{3}\int_{\sqrt{1+z^2}}^{\sqrt{9-z^2}}rdrdzd\...
- 27
1
vote
2
answers
72
views
Find the volume of the solid obtained by rotating the region bounded by the graphs $y=\frac{1}{x}, y=0, x=1$ and $x=9$ about $ y=6 $
Find the volume of the solid obtained by rotating the region bounded by the graphs $y=\frac1x,y=0,x=1$ and $x=9$ about $y=6$.
I thought that maybe by offsetting $\frac{1}{x}$ by 6 and then treating it ...
- 25
0
votes
0
answers
12
views
Calculate a box closest to some limit that fits in a larger box
I'm not sure how to summarize this problem in the title and see a bunch of seemingly related Q&As but I didn't find one that fits this particular issue.
I have box with limits in 3 dimensions (...
- 101
1
vote
2
answers
37
views
Calculate the volume $λ_3(A)$ of $A := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 ≤ 1\} $
I have a lot of problems with exercises where I must calculate the Volume of a set using integrals.
Here an example:
Le the set $A$ be $A := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 ≤ 1\} $. ...
- 432
2
votes
1
answer
71
views
Whats wrong with my solution in this calc problem?
The question asks me to
Find the volume common to two spheres, each with radius $r$, if the center of each sphere lies on the surface of the other sphere.
I've seen the same question asked in other ...
- 235
0
votes
0
answers
29
views
How to find the points of intersection of two multivariable functions on their outer bounds?
I have two multivariable functions, and I want to calculate where they intersect on the outer bounds of $x$ and $y$ between the graphs of $f(x,y)$ and $g(x,y)$.
The functions are:
$f(x,y)=x^2+4y^6$
$g(...
1
vote
1
answer
26
views
Finding the bounds of a multivariable integration to compute the volume between two functions?
I have two multivariable functions and I want to find the bounds to integrate.
$f(x,y)=x^2+4y^6$
$g(x,y)=\sin{x}+\frac{1}{5}y+4$
How can I determine the bounds to integrate. Thereby, I want to ...
- 11
1
vote
1
answer
36
views
How to apply the volume of solid revolution formula in this case?
I'm trying to find the volume of a object using the solid revolution formula:
$$\int_a^b \frac{\pi}{2}y^2 \;\mathrm{dx}$$
The shape of my object can be given by this equation:
$$(x^2+y^2)^2=4x^3-2xy^2$...
- 303
2
votes
0
answers
41
views
Volume of half an ellipsoid
I want to calculate the volume of the solid defined as follows.
$$ K := \left\{ (x,y,z) \in \Bbb R^3 : x^2 + y^2 + 3z^2 \le 4, z > 0 \right\} $$
I know that this is a half solid ellipsoid and that ...
- 379
1
vote
0
answers
27
views
Isoperimetric theorem for tetrahedra, hexahedra and dodecahedra
Note: This is an optimalization problem. You can scroll to below to see what needs to be optimalized
I am trying to prove the isoperimetric theorem for dodecahedra, hehahedra and dodecahedra, the ...
- 1,123
0
votes
0
answers
30
views
Find the volume - using substitution
Find the volume of $\omega=\{(x,y,z) | x^2+y^2+z^2\leq a^2 , x^2+y^2\leq ax\}$
$x^2+y^2+z^2\leq a^2 \implies x^2+y^2\leq a^2-z^2$.
Therefore , $a^2-z^2 \leq ax \implies a^2-ax\leq z^2$.
Using ...
- 2,149
1
vote
1
answer
41
views
How to fine the formula for the volume using cylindrical shells?
Use cylindrical shells to find the volume of the solid. The region bounded $y = x^2$ and $y = 1$ is revolved around the line $y = 3$.
The answer is:
$$\int_0^1 4\pi(3-y)y^{1/2}dy$$
Why is it $4\pi$? I ...
- 11
1
vote
1
answer
68
views
The volume obtained by rotating the region bounded by $y=\sqrt{81-(x-4)^2}$ and the $x\ -$ axis about the line $x=14$.
Question: The volume obtained by rotating the region bounded by $y=\sqrt{81-(x-4)^2}$ and the $x\ -$ axis about the line $x=14$.
I used both Disk and cylindrical shells method to evaluate the volume. ...
- 936
8
votes
5
answers
373
views
Why is the volume of a cone not half the volume of the cylinder of which it is a part?
Now I'm aware of the proof for the volume of a cone being a third of the volume of the corresponding cylinder and the proof is satisfactory.
Here's an alternate approach that seems to imply that the ...
- 97
3
votes
2
answers
62
views
>Find the volume of the solid generated by revolving about the y-axis the region bounded by the curve $y=x^3$ and the lines $y=0$ and $x=2$
Find the volume of the solid generated by revolving about the y-axis the region bounded by the curve $y=x^3$ and the lines $y=0$ and $x=2$
My attempt: Take radius $r=(2-y^{1/3})$
Then volume $V=\int_{...
- 1,970
0
votes
0
answers
42
views
What is the volume of the cuboid?
A cardboard box company uses a machine that creates cuboid (rectangular prism) boxes with outer dimensions (meaning the dimensions of the box if you are measuring from the outside of the box) of $(x+...
- 539
0
votes
0
answers
32
views
Gauss' formula on positively oriented closed contour
Show for a positively oriented simple closed contour $C$ that the area of the region $G \subset{\mathbb{C}}$ enclosed by C is given by $\frac{1}{2i}\int_C \bar{z}\text{ d}z$. Use the Gauss' integral ...
- 1
1
vote
1
answer
50
views
Finding the volume between cylinders, paraboloid and plane
I need to find the volume of the solid between the cylinders
$$x^{2}+y^{2}=1,\quad x^{2}+y^{2}=4$$
and limited by
$$ z = 0, \quad z = 9 - x^2 + y^2 $$
i got this far but i dont know if it is right ...
- 25
1
vote
2
answers
28
views
How count numbers of containers in filing containers of three different sizes with known exchange rates using algebra rather than via my shortcut?
A manufacturer packages soap powder in containers of three different sizes. The amount of soap powder in a full large container could fill exactly 3 of the medium containers or exactly 5 of the small ...
user685056
1
vote
0
answers
28
views
Question about ratio of Riemannian densities
I am studying the paper "On the MCP property of metric measure spaces" by S. Ohta (https://ems.press/content/serial-article-files/1508) and I have a question regarding a passage at page 813 ...
- 429
0
votes
1
answer
32
views
The volume of a 3N-dimensional hypershell
R.K. Pathria in his book Statistical Mechanics has written that the volume of a $3N$-dimensional hypershell, bounded by hyper spheres of radii $$\sqrt{2m(E+\tfrac12\Delta)}\; \text{and}\; \sqrt{2m(E-\...
- 2,585
3
votes
1
answer
30
views
Find the volume of the solid obtained by rotating the region bounded by the given curves around the x-axis
The region bound by $y=x^2, y=-x, y=3$, rotated about $x$-axis. I tried to solve it by first finding the volume of the region bounded by $y=x^2$ and $y=3$ ($\frac{36\sqrt{3}}{5}\pi$). Then I found ...
- 33
0
votes
1
answer
27
views
For a solid of revolution, how do I determine what expression I should use for my radius when the axis of rotation has been shifted from the axis?
For instance, consider the following case:
The region of the $xy$-plane bounded by the $x$-axis, the line $x = 2$, and the graph of $y = x$ is rotated about the line $x = 4$ to construct a solid $S$. ...
- 335
1
vote
0
answers
27
views
Volume under sphere bounded by cylinder and plane $z=0$
Hello, I've tried calculating the volume under the sphere using these integrating bounds but it gives me the wrong answer.
- 329
0
votes
0
answers
22
views
How to calculate volume of root lattices $A_n$
Is there anywhere in the bibliography an explanation as to why the volume of the root lattice $A_n$ is $\sqrt{n+1}$?
$$A_n = \biggl\{(x_0,x_1,\dots,x_n) \in \mathbb{Z}^{n+1} : \sum_{i=0}^{n} x_i = 0\...
- 179
2
votes
0
answers
48
views
Is Gabriel's Horn possible in four dimensions? [duplicate]
Gabriel's horn, a shape with infinite surface area but finite volume, is one of the most counterintuitive objects in mathematics. However, it is pretty intuitive that we can build a Gabriel's horn in ...
- 3,873
0
votes
1
answer
27
views
Proportion of regular tetrahedron occupied by mutually tangent balls centred at its vertices
The centres of four balls of radius $1$ are the vertices of a regular tetrahedron of side length $2$. What is the proportion of the tetrahedron occupied by the balls?
At first I thought it should just ...
- 8,368
1
vote
1
answer
53
views
Why does $\int_0^s \frac{\sqrt{3}}{4}x^2 dx$ not give the volume of a regular tetrahedron with side length $s$?
Suppose I want to find the volume of a regular tetrahedron with side length $s$. Intuitively, I thought we could sum over the areas of all equilateral triangles of side length $x$, where $0 \leq x \...
- 2,775
0
votes
1
answer
46
views
Uncertainty when calculating volume of a rectangular block
I'm trying to determine the uncertainty in volume , $ V = b \cdot h \cdot l$ , of a rectangular block with the following measurements,
$$ b \quad=\quad 2 \pm 0.1 $$
$$ h \quad=\quad 2 \pm 0.2 $$
$$ ...
- 473
0
votes
2
answers
82
views
Calculating the volume between 2 surfaces
I have to calculate the volume between $z = 1+x^2+4y^2$ and $z^2=4x^2+16y^2$ in the half plane $z \le2$.
I have set the bounds as $ -\sqrt{\frac{1}{4}} \le y \le\sqrt{\frac{1}{4}} $ and $-\sqrt{-4y^2+...
- 27
2
votes
2
answers
127
views
The volume of an $n$-dimensional unit hypercube intersected with the plane $\sum_{i=1}^n x_i=0$.
Let $\mathcal{C}$ be the unit cube in $\mathbb{R}^n$ centred at $\mathbf{0}$, for some integer $n>1$. Consider the following $(n-1)$-dimensional subspace:
$$
V=\left\{(x_1,x_2,\dots,x_n) \in \...
- 658
2
votes
2
answers
84
views
What's the difference between the ellipsoid described by $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ vs $<1$?
I have to find the volume of the ellipsoid described by the set $ E = \{(x,y,z) \in \mathbb{R}^3 |\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}<1\}$. I have a few ideas and there is a bit of ...
- 67
0
votes
1
answer
28
views
Find the volume of ball using variable substitution
$B = \left\lbrace \,
(x, \; y, \; z) \;
\left| \; \; \;
\rule{0pt}{12pt}
\right.
(3 x + 3 y + z)^2 \;+\;
(3 x + y + 3 z)^2 \;+ \;
( x + 3 y + ...
- 2,149
0
votes
0
answers
14
views
Limits of r and rho, projection or not?
When taking triple integrals and converting cartesian coordinates into cylindrical or spherical coordinates, are the bound of $\rho$ determined by the distance from the origin of the $3$ dimensional ...
- 1
1
vote
0
answers
54
views
Tensor of inertia of a curved beam
I need help calculating the inertia tensor for a curved beam.
I found the formulas for this in the article https://hal.archives-ouvertes.fr/hal-01084693/document[page 20, Appendix A] and decided to ...
- 681