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Questions tagged [volume]

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

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Using disk method in integration

I need to find the volume of the region defined by the inequalities $$\begin{align*}y & <\frac 12\\0< y & <1-x^2\end{align*}$$revolved around the $x$ axis. I tried separating them and ...
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$V(T(E)) \geq c^n V(E)$

$T:\mathbb R^n \to \mathbb R^n$ is a diffeomorphism such that $\forall x,y \in \mathbb R^n: |f(x)-f(y)| \geq c|x-y|$ $E \subset \mathbb R^n$ is Jordan measurable. Show that $V(T(E)) \geq c^n V(E)$ ...
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Sphere inscribed in a cone

If a cone of height h and radius r has a sphere inscribed in it such that it touches the base and the curved surface area, how can I find the radius of the sphere? (Is this in the level of a 9 grader?)...
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$\lim_{k \to \infty}v(E_k) = v(E)$

$E_1,E_2, \dots$ are measurable sets, $E \subset E_i$ for all $i$ is also a measurable set and bounded. In addition we are given that for all $k \in \mathbb N$ and for all $x \in E_k$ there is a $y \...
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how to calculate wall thickness of a mesh?

based on similar questions on mesh volume, volume of a mesh can be calculated by following equation: volume = ((vec1 x vec2) . vec3) /6 where vec1, vec2, and vec3 are the vectors from origin to a ...
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What is the water level of an oblique bottle

If a bottle, perfectly cylindrical, has a radius r cm and height h cm, the water level is l. Then if the bottle is made oblique, what would be the new water level from the point of contact with the ...
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1answer
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Volume of liquid in a hopper / inverted frustum of a pyramid, based on the liquid level.

This question is part of a larger investigation into specific machinery process rates. I know the dimensions of the hopper, and thus the volume of the hopper (inverted frustum of a pyramid). However, ...
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volume between a sphere and a cone with spherecoordinates

$D$ is the surface under the sphere with radius $r=1$ and center $M(0,0,1)$, and above the cone defined by the equation $z=\sqrt{x^2+y^2}$. The question is to find $D$ with sphere coordinates. And to ...
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2answers
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Volume of a solid using disk method

I have the following homework problem which I can't seem to get right. The question stated is: Find the volume of the solid generated by revolving the region bounded by the parabola $y=\frac{x^2}...
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How to calculate the area and volume of a random 3d shape knowing only its coordinates?

Given a set of 3d points which make up an vector object of any shape (with any number of points), without the edges being known, how can the object's edges be found/detected so that the object's ...
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what is the interval for this cylindrical shell problem?

For the question attached, my textbook integrates from the interval 0 to 1, but shouldn't it be integrated from the interval of -1 to 1?
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1answer
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Find $\lambda^{d}(A_{d}(a))$ and determine its behaviour as $d \to \infty$

Set $A_{d}(a):=\{(x_{1},...,x_{d})\in\mathbb [0,\infty[^{d}: \sum_{i=1}^{d}x_{i}\leq a\}$, whereby $a > 0$ Determine: (i) $\lambda^{d}(A_{d}(a))$ (ii) how $(A_{d}(1))$ behaves as $d \to \infty$ ...
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Bound on volume of $A-B$ according to Minkowski

$A-B:=\{c:B+c\subseteq A\}$, in, say, Euclidean space. I think that if $A-B$ is not the universe, then $vol(A-B)\leq vol(A)$ (If $B$ has one point then the inequality is immediate, adding more ...
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1answer
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Having difficulty understanding how to find the maximum volume with cost constraint.

We have 1000USD to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides costs 10USD/cm2 and the material of the bottom costs ...
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1answer
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Volume of intersection of a sphere and a paraboloid

Could I calculate the volume of the intersection of $x^2+y^2+z^2=8$ and $4z=x^2+y^2+4$ using spherical coordinates and treating the paraboloid as a function, as in the following example? $f(x,y)=\...
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What is the volume of the region $S =\{(x, y, z) : |x| + |y| + |z| ≤ 1\}$?

What is the volume of the region $S =\{(x, y, z) : |x| + |y| + |z| ≤ 1\}$ ? How can i find the volume any hints /solution
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1answer
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Rotational volume and differential equation

A container with the shape of some function y=f(x) is rotated around the y-axis. It's filled with a fluid and has a hole in the bottom where the fluid leaks out. The rate of liquid flowing out should ...
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1answer
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Simplest Way to Evaluate Lengthy Integration Equation with Succinct Answer

Recently, I determined the volume of the solid of revolution created by rotating the parabola, $y=ax^2$, about the axis of revolution, $y=bx$, as a function of $a$ and $b$, where both variables are ...
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Volume of a rectangular prism after t days

I'm really stuck on this question and would like some direction. An iceberg approximates the shape of a rectangular prism and is about 800 m long, 400 m wide and 120 m thick. As it moves into warmer ...
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1answer
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What is the simplest way of getting the solid angle $\Omega_d$ in a space of $d$ dimensions?

It is known that the solid angle in a flat space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$) is given by these formulae: \begin{align}\tag{1} \Omega_{2 n} &= \frac{1}{(n - 1)!} \, 2 \pi^n, \...
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a question on the volume of a set

Can anybody give me a hint on this? Let $v\in \mathbb{R}^n,\quad r,R,C$ positive constants . We would like to calculate the volume of the set $$A=\{w\in \mathbb{R}^n : r\leq |w| \leq R, |w-v/2|\leq C \...
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The volume of solid of revolution rotated about the line $y=x$ [closed]

Find the volume of solid of revolution of region between curves $y=\sqrt x$ and $y=x^2$ in $xy-$plane about the line $y=x$. I know the answer, $\pi/30\sqrt 2$, but how we can obtain it? Should we ...
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1answer
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The area of a truncated pyramid with irregular top and bottom surface, given the height $h$

This question is inspired by the question The volume for truncated pyramid with irregular base. Given that we have the top and bottom surface area ($A_1$, $A_2$) of a pyramid, ad the height of the ...
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3answers
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Compute the volume of a solid by revolving a region about the $y$-axis

How to compute the volume of the solid generated by revolving the region between the curve $y=\dfrac{\cos x}{x}$ and the $x$-axis for $\pi/6\leq x\leq \pi/2$ about the $y$-axis? I think we need to ...
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Volume of an Irregular Octahedron from edge lengths?

Does anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. ...
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Center of mass of an object in $\mathbb{R^3}$

A few days ago I asked this question. I have solved what I was asking there. The mass of the given object is $$M=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{\pi/3}\displaystyle\int_{0}^{1}k\...
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2answers
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Find volume with triple integral

I want to find the volume of $ (x^2 + y^2)^3 + z^6 =a^3xyz $. I changed the basis like that: $ x = r \sin ^ \frac{1}{3} B \cos A; y = r \sin ^ \frac{1}{3} B \sin A; z = r \cos ^ \frac{1}{3} B $ (...
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Volume of Solid of Revolution (Glass)

We are given $y^2/a^2-x^2/b^2=1$, $y>0$ . If we rotate the hyperbola around the $y$ axis the shape is similar to a glass. What will the volume of water inside the glass be, in order to fill the ...
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1answer
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Volume of solid of revolution ($\cos(x)$)

Compute the volume of the solid of revolution that results from revolving $f(x)=\cos(x)$ between $x=-\pi/2$ and $x=\pi/2$ around $y=-1$. I know how to do so around the $x$-axis or the $y$-axis, ...
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1answer
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How to Solve It: A Rate Problem

I am currently reading How to Solve It by G. Polya. I am confused by the conclusion in one of his examples. You can find it here in this pdf on page 29. The problems states: "Water is flowing into a ...
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Simplest Way to Find Volume of Solid of Revolution Around Given Line

Question I would like to know the simplest way to find the volume of the solid of revolution created by rotating the parabola $y=x^2$ around the line $y=x$ (the shape shown in blue below). I am ...
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How do I find the volume of a parallelepiped given 4 vertices?

"Find the volume of the parallelepiped by four vertices: $(0,1,0), (2,2,2), (0,3,0),$ and $(3,1,2)$. I know the formula to find this volume is: $|\vec{a} \circ(\vec{b}\times \vec{c})|$, and I know ...
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1answer
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Find the volume bounded by $C$ and the planes ( integrals ) ( question 9 )

Let $C$ be the cylinder $\big\{(x,y,z):x^2+y^2\leq 1\big\}$. Find the volume bounded by $C$ above the plane $z=0$ and below the plane $x+z=1$. I have wondered how to solve this , here is my attempt :...
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Find the volume of the solid generated by revolving the region bounded by $y=\sqrt{9-x^2}$ and $y=0$ about the x-axis.

The answer for this question is apparently $36\pi$, but I cannot get that answer. I keep getting $18\pi$, and I have a feeling my professor made a mistake on this. Can I get some clarification?
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3answers
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volume of surface given by $(x^2+y^2+z^2)^2=x$

A question asks me to find the volume of the surface $(x^2+y^2+z^2)^2=x$ this looks like a very difficult triple integral to evaluate using cartesian coordinates, so I though I would describe the set ...
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Comparison of volume forms on Riemannian manifolds.

I am reading the Cheng's paper (1975), which states that Theorem. Suppose $M$ is a complete Riemannian manifold and Ricci curvature of $M\geq(n-1)k,\ n=\mathrm{dim}\ M.$ Then, for $x_0\in M$ we have $...
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How to find volume and surface area of a spindle torus?

I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found ...
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getting volume with given region and bounds

So in my text book(online) we have this question and I'm very lost what to do as my answer is wrong. The region bounded by the curve $y=x^2$ and the x-axis between $x = 1$ and $x = 0$ is revolved ...
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Proof of volume of spheres through sum of sections of a sphere [duplicate]

I know that by integrating the surface area of a sphere we can get the formula of its volume. Similarly, I wanted to find about other ways of doing this. So, I thought what if we divide the sphere ...
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Given the cylinder $(x-1)^2+y^2 \leq1$ and a sphere centred at $(0,0,0)$ with radius $2$, find the volume of the solid formed.

I attempted this using cylindrical co-ordinates but I am not sure if my attempt is correct. This is what I have: equation of sphere : $x^2+y^2+z^2=4,$ cylindrical coordinates this is $z^2=4-r^2$ ...
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Find the volume of the part of the vessel under water

I am struggling to find the volume of the part of the vessel that is partially under water in the third figure (the one where the elephant is on top). Is there anything missing from the given ...
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Calculate volume with Cavalieri's principle for $4x^2 + 0,25y^2 \leq 1$

Let there be the Ellipse When you rotate $E$ around the x-axis, then one gets the Spheroid $R$. How can I calculate the volume of $R$ with Cavalieri's principle? I used which is equivalent to ...
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Triple integral - switching limits around

I have been attempting this question for more than four hours now. Since I don't have the answer key for it, I try and check my answer using Symbolab by ensuring all three integrals give the same ...
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1answer
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Number of hypercubes that fit into a hypersphere with radius 1 and centered at the origin

There is a hypercube with sides of length 2 and a hypersphere with a radius of 1. Both of them are centered at the origin (the hypercube surrounds the hypersphere). Each of the d dimensions in the ...
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prove ratio between hypervolume and surface volume of n-sphere is optimal

The ratio of the surface volume to hypervolume of an $n$-sphere is $\frac{r}{(n+1)}$ . Is there a way to prove that this is the highest ratio possible without appealing to the fact that $n$-spheres ...
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1answer
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Volume with double integral

Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$. From this, I got that the volume would simply consist of the region under $z = 1-x-(5/6)y$. And as $z=0$, ...
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Finding the height and top radius of cone so that volume is maximum & Finding the angle so that the volume is maximum

You have a 6-inch diameter circle of paper which you want to form into a drinking cup by removing a pie-shaped wedge with central angle theta and forming the remaining paper into a cone. - You are ...
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How to calculate atypical object volume

I am not a mathematician but a simple programmer and came across a situation where I have to calculate different kinds of object volume. There is no specific type of object and i found that there are ...
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1answer
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Volume of Water Inside a Cup

I came across this problem in School. Suppose I have a cup and suppose that the cup is cylinder. I know the total volume of the cup and I fill with some water. Let $V_c$ be the volume of the cup, $...
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Volume of ellipsoid proportional to inverse of matrix

In a book on convex optimization it is said that if an ellipsoid is defined as $$\mathcal{E} =\{v:\|Av-b\|_2\le 1\}$$ with $A\in S^n, A \succ 0$ ($A$ is a positive definite matrix and $b$ is a real ...