For questions related to volume.

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0
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1answer
15 views

volume using spherical coordinates

Let $$V = \{(x, y, z): x^2 + y^2 ≤ 4 , 0 ≤ z ≤ 4\}$$ be a cylinder and let $P$ be the plane through $(4, 0, 2), (0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I'm having ...
5
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1answer
41 views

Compare determinants of matrices with different dimensions

Reading about matrices and determinants I am wondering about the following concept: How valid is to compare the determinants of matrices with different dimensions? e.g. compare a determinant $D1$ ...
0
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1answer
39 views

How to calculate the volume of the solid described $x^2+ y^2+z^2 \le 9$ and $2 \le z \le \sqrt5$?

How to calculate the volume of the solid described $x^2+ y^2+z^2 \le 9$ and $2 \le z \le \sqrt5$? I try $x=2r \cos \phi$, $y=2r \sin \phi$, $z=z$, but but probably not the way to go
2
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0answers
27 views

Using Gauss's theorem to determine the volume

I'm currently stuck with this kind of problem: Let $S$ be the surface in $\mathbb{R}^3$ obtained by evolving the curve $x = \cos u$ and $z = \sin 2u$ where $-\frac{\pi}{2} \leq u \leq ...
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0answers
30 views

Cylindrical/ Spherical Coordinates Integral - Volume of Cone

"(b) Let $C$ be the solid cone with the boundary surfaces $x^2 +y^2 = z^2$ and $z = 0$. The density of the solid at point $(x, y,z)$ is $z$. Find the volume of the solid using the integrals in both ...
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2answers
26 views

Matlab Monte Carlo Method Volume Calculation

The question: use the Monte Carlo method to find the volume of intersection of, three cylinders, all of radius 3 units and infinite in length,with the axis of the first cylinder being the x-axis, the ...
0
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1answer
22 views

How can I find the volume of a solid bounded by these functions?

The region bounded by $x^2 - y = 0$ and $x+y=0$ is rotated around $y=0$ I've drawn a graph of each of these but I'm just not getting it conceptually. The biggest problem I have is knowing when to use ...
2
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2answers
36 views

How do I prove that the area of a sphere is the least possible area for a given volume?

Or, why do soap bubbles have the shape of a sphere and not that of unicorns? What's the math behind it?
15
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1answer
72 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...
2
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0answers
28 views

What is the volume inside $S$, which is the surface given by the level set $\{ (x,y,z): x^2 + xy + y^2 + z^2 =1 \}$?

The solution given uses a linear algebraic argument that doesn't seem very instructive -- and may not even be correct, I think. We notice from the equation, that the surface is a quadratic form, ...
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2answers
46 views

A hole of radius 1 is drilled to the centre of a ball of radius 2

A hole of radius $1$ is drilled into the centre of a sphere of radius $2$. Calculate the remaining volume. Could someone explain to me how to approach this. I don't need a specific answer. I know ...
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1answer
28 views

connected rates of change - help!!! [closed]

I really would appreciate your help with this question. I got stuck on it whilst revising for my c4 exam.
1
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1answer
42 views

Calculating the volume of a cylinder.

Let $V = \{(x, y, z): x^2 + y^2 ≤ 4$ and $0 \le z \le 4\}$ be a cylinder and let $P$ be the plane through $(4, 0, 2)$, $(0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I ...
0
votes
1answer
11 views

Does a bounded countably infinite union of sets with volume have volume?

If $ A_1, A_2,...$ are sets with volume and $A= \cup_{i=1}^\infty A_i$ is a bounded set, must $A$ have volume? This was a homework problem that we went over in class, and if I remember correctly the ...
0
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1answer
37 views

Cavalieri’s Principle for calculating volume.

Let $B = \{(x, y, z): x^2 +y^2 +z^2 ≤ 4\}$ be the ball with radius $2$ in $\mathbb{R}^3$ and let $V$ be the region inside $B$ above the plane $z = 1$. Use Cavalieri’s Principle to compute the volume ...
2
votes
1answer
34 views

How to calculate the volume of a skip bin container knowing the height of the material inside

I need to know hot to calculate the volume of a skip bin (also known as a skip container or dumpster in some areas) with varying length and width. It seems like a isosceles trapezoid when you look at ...
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0answers
19 views

Difference between measure zero and volume zero?

I have the following definitions for a set to have measure zero and for a set to have volume, respectively: A set $A$ has measure zero if for any $\epsilon > 0$ there is a covering $\{S_i\}_{i \in ...
0
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1answer
30 views

Integration: Finding area, volume and arc length

I am new to integration, so please do not mark this question as "not enough research done" Here is the question (please open image in new tab to see it clearly) - I am getting stuck with the ...
1
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0answers
19 views

Volumes of solids (of rotation). Any real world applications?

Washer method, disc method, etc. In what areas or fields would someone make these calculations? For example, I think 3D printers do some sort of "slicing" algorithm in their CPU in order to print ...
1
vote
1answer
26 views

displacement of water

I just read the following puzzle on puzzling SE and it made me wonder, is there a proof which can be formulated to see if an object of any size and weight would cause more displacement inside the boat ...
5
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1answer
60 views

Volume of a hole through cylinder (from the side)

I need to calculate the volume of a circular hole in a cylinder and I've come across a problem. The problem is finding the "cap-volume", which is needed to complete the volume of the hole. I created a ...
2
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0answers
30 views

Perimeters Areas and Volumes

I have to write an article for a school magazine. I thought it is better to choose a simple topic like Perimeter, Area and Volume. I am looking for historical fact and surprising facts about ...
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2answers
47 views

Volume of solid by Spherical

Trouble setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. Use integration with Spherical coordinates. (Hint: Use two ...
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1answer
41 views

Triple Integrals

Find the volume of the region bounded by $(x^2+y^2+z^2)^2=x.$ I'm having issues setting my bounds (specifically $\theta$) so far I have $0<r< [\sin(\phi) \cos(\theta)] ^{(\frac{1}{3})}$ ...
1
vote
1answer
42 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
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0answers
9 views

Volume of a region given by a CSP

I have a Linear Constraint Satisfaction Problem i.e. I have variables $ x_1, x_2,...,x_m$, with corresponding domains $D_1,D_2,...,D_m $ satisfying linear constraints $C_1, C_2,...,C_n$ with $n ...
0
votes
2answers
24 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
0
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1answer
43 views

Calculate the Volume of the unitary n-ball

To do this first I need to prove that : $$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1 \cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$ where $ ...
0
votes
3answers
25 views

A $1.00 m^3$ volume of iron has a mass of $7.86 \cdot 10^3 kg$, find the length of one side with a mass of $200g$

So this is a relatively simple physics problem, but I'm sort of stumped and need some help finishing this off. The full problem reads: Iron has a property such that a $1.00 m^3$ volume has a mass ...
0
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1answer
20 views

What does this volume represent?

I have been trying to draw this out for an hour now and cannot visualize it. $x$ is between $0$ and $1$, $y$ is between $0$ and $x$, and $z$ is between $x^{2}+y^{2}$. The $z$ line is just a ...
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0answers
23 views

Solids of Revolution around other functions.

Recently I've been thinking about solids of revlution, and thought about an interesting experiment. Can you rotate functions around, for example, the line $f(x)=x$? And consequently, could you rotate ...
3
votes
1answer
79 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
3
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0answers
27 views

How to calculate a reduced volume?

Let's say we have an irregular 3D shape with volume=V ( we know V but we don't know its equation= F). Now I want to calculate another 3D shape which is exactly the same shape but one size smaller, ...
0
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2answers
32 views

Volume of a region?

It is my intuition that the volume of the solid such that $0\leq x_1 \leq x_2 \ldots \leq x_n \leq T$ is $\frac{T^n}{n!}$. Can someone confirm/deny and/or supply proof? Thanks!
0
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0answers
17 views

Convex pyramid with same pyramid volume formula

Areas $ A_1,A_2 $ of parallel planes are of $n$ sided polygons spaced distance/ height $H$ apart. How should generators of a solid be defined so that solid volume can continue to be calculated by the ...
4
votes
1answer
36 views

Deriving the formula for volume of a cone - middle school

I realize there are several solutions for calculus-based proofs, but how would you explain the formula for the volume of a cone to middle school students? Please do not give the un-rigorous "imagine ...
1
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1answer
25 views

Parametrization of this surface

Consider the following surface: $$A=\{(x,y,z,t)\in\mathbb{R^4}:0\leq x,y,z,t\leq1,\quad x+y+z+t=1\}$$ I need to parametrize it to be able to calculate its volume. Of course, I thought on seeing it ...
1
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1answer
55 views

Is the area of $T(\Omega)=|\det A|\,(\text{area of }\Omega)$?

We are given that $\Omega$ is a parallelogram in $\mathbb{R}^3$ and $\left\{ T(\vec{x}) = A\vec{x} \mid \mathbb{R}^3 \mapsto \mathbb{R}^3\right\}$ is a linear transformation. From the definition of ...
2
votes
1answer
72 views

What is the 3-volume of the 3-parallelepiped defined by $\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}$?

We have $\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}=\left\{\begin{bmatrix}1\\0\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\2\\3\\4\end{bmatrix}\right\}$ ...
2
votes
2answers
34 views

Determining the rate of change of a radius as a sphere loses volume

Problem: A spherical balloon leaks $0.2\mathrm m^3 / \mathrm{min}$. How fast does the radius of the balloon decrease the moment the radius is $0.5\mathrm m$? My progress: Since we're dealing with ...
0
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0answers
33 views

Find the Volume Contained Inside a Sphere and Cylinder

Find the volume of the region that lies inside the cylinder $x^2 + y^2 = 1$ and $x^2 + y^2 + z^2 = 4.$ I've attempted to break this down into 2 sections: a pure cylinder, which continues until it ...
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0answers
21 views

Question Regarding Double vs Triple Integrals

I'm having a very difficult time understanding how double and triple integrals are both used to represent 3-dimensional space, and how you decide which method is used to evaluate volume. Is anyone ...
0
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1answer
51 views

Interesting trig integration pattern?

I was recently playing around with some easy volume by revolution problems that I just randomly make up for fun, and I found a weird and interesting pattern that I can't easily (or otherwise) explain. ...
3
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1answer
110 views

Why does this sphere volume proof work?

See this video, but I can explain it without you watching it (nothing important is said and you can see the video in mute): Basically all it does is cut the sphere into 'infinitesimal' pyramids ...
0
votes
1answer
48 views

Find the volume of the solid bounded by surfaces.

Find the volume of a body formed by rotation around an axis $ V_{x}$ Figure bounded by the graphs of functions: $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$ $$x^{2}-\frac{y^{2}}{15}=1$$ $$x \geq 1$$ ...
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4answers
51 views

Volume formed by rotating square by 360 deg

If I know the area of the square and I rotate it around one of the edges by $360$ deg, it forms a solid cylinder. Why then is the volume of this cylinder not equal to "area of the Square" multiplied ...
0
votes
2answers
33 views

Evaluate the integral over the region R

I am a bit lost on how to evaluate double integrals over a region. I am asked to evaluate the following integral $$\iint\frac{y}{(x^2+y^2} dA$$ over the region R: triangle bounded by $y=x, y=2x, ...
2
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1answer
19 views

Use the method of cylindrical shells to find the volume generated by rotating the region bounded…

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves $y=x^2, y=2-x^2$ about the line $x=1$. I'm just trying to set this one up. Is this ...
0
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1answer
27 views

volume between a sphere and cone

I'm having some problems finding the volume between a sphere of radius 5 and the cone $z = -\sqrt{x^2 + y^2}$. the bounds I got for spherical coordinates are 0 to 2pi for theta, 3pi/4 to pi for phi, ...
2
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0answers
73 views

Question about Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies - Kannan Lovasz Simonovits 97

I've been trying to understand this paper: "Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies", Ravi Kannan, Laszlo Lovasz, Miklos Simonovits. Motivation: The paper is about ...