Questions tagged [volume]

Question 1

Here is the cross-section of an ellipsoid that has rotational symmetry around $b$. It approximates a pinned droplet on a smooth surface (pinned meaning that its contact area is constant while the ...

Question 2

Problem In three-dimensional $xyz$-space, consider the cylindrical surface given by $x^2+y^2=1$, and let $S$ be its portion with $0\le z\le 2$. A sheet of paper of negligible thickness is wrapped ...

Question 3

I want to find volume of shape on picture below. Red vector is $\vec{a}$, green is $\vec{b}$ and blue is $\vec{c}$. Vectors are right-handed and located in first coordinate octant. I've suggested that ...

Question 4

How this shape called and why this is not a pyramid? Here link to Desmos3D. I wanted to find a volume of shape on 3D-vector $ v=(v_1, v_2, v_3) $ and thought it's a pyramid with volume $ \frac{1}{3} |...

Question 5

2D case Let a convex quadrilateral $Q= \operatorname{conv}\{A_1,A_1',A_2,A_2'\}$ have vertex pairs $$(A_1,A_1'),\quad (A_2,A_2'),$$ and define the “diagonal vectors” $$d_i = \overrightarrow{A_iA_i'} = ...

Question 6

I would like to calculate the volume of the section shown in blue in the 2 drawings below, which section results from the collision of two liquid jets. The base for the rotation is an ellipse as shown ...

Question 7

I have a question regarding polytopes. I have a constraint matrix and a vector. This matrix has a set amout of equalities on top and then the rest is inequalities. I know the number of equalities. As ...

Question 8

I am reading Analysis on Manifolds by James R. Munkres. On pp.168-169: Exercise 6. Let $B^n(a)$ denote the closed ball of radius $a$ in $\mathbb{R}^n$, centered at $0$. (a) Show that $$v(B^n(a))=\...

Question 9

I have three parametric equations in two variables that give the coordinates of points on a three-dimensional, closed, convex surface. I want to find the volume enclosed by that surface, but I haven't ...

Question 10

I am reading "Analysis on Manifolds" by James R. Munkres. Definition. Let $S$ be a bounded set in $\mathbb{R}^n$. If the constant function $1$ is integrable over $S$, we say that $S$ is ...

Question 11

We have a narrow six-sided prism with a height of $50\space\text{cm}$. Also, the base has sides of $\sqrt[4]{3}\space\text{cm}$ each (a regular hexagon). We add water up to a height of $40\space\text{...

Question 12

Suppose $S\subset \mathbb{R}^n$ is measurable and has $\text{Vol}(S)$. If $M$ is a linear transformation then it is known that $\text{Vol}(MS)$ is equal to $\text{Vol}(S)$ multiplied by the product of ...

Question 13

I need to calculate the volume (and centroid, but techniques for both seem to be fairly similar) of the intersection between the unit cube defined by $0 \leq x,y,z \leq 1$ and the halfspace defined by ...

Question 14

This is probably very simple, but I can't remember. In $\mathbb{R}^n$, the "signed area" of a parallelogram by vectors $v_1, v_2$ is defined as $v_1\wedge v_2:=|v_1\wedge v_2|n$, where $n$ ...

Question 15

I need help understanding how to work out the following: You have a cake that is 8 cm in diameter. By what factor would you need to increase the diameter of this cake to double the volume (assuming ...

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Questions tagged [volume]

Angle at base of ellipsoid cap for fitting data

Volume of the cylinder surface

How to find volume of solid body which is formed by right-handed triplet of vectors in first coordinate octant?

How to find volume of solid body built on 3-dimensional vector?

Volume of polytope spanned by vertex pairs $(A_i,A_i')$ is invariant under translation of diagonals $A_iA_i'$

Calculate the volume of a variable trapezoid rotating around an axis

Parametrization of Polytope to Make it Full Dimensional

$\lambda_n$ is the volume of the $n$-dimensional unit ball. Compute $\lambda_n$ in terms of $\lambda_{n-2}$. (Analysis on Manifolds James R. Munkres.)

Is it possible to calculate the volume of a general 3D (closed & convex) parametric surface?

Why do we assume that $S$ is rectifiable and then consider the integral of a bounded continuous function $f$ over $S$? Analysis on Manifolds Munkres..

Finding the distance between water surface and the wall of prism

Proof (or reference) that the measure of a set $S\subset\mathbb{R}^n$ is preserved by an orthogonal transformation $V$?

Volume/centroid of unit cube cut by plane

Some confusion over exterior algebra and signed area/volume

Calculating difference in diameter for cylander that scales [closed]

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