Questions tagged [volume]
For questions related to volume, the amount of space that a substance or object occupies.
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Transformation of contraints given non-invertible transformation of variables
Given the transformation $Ax = y$ and constraints $Cx \le d$, how to obtain the resulting constraints on $y$ when $A$ is rectangular with unknown rank? I'm thinking along the lines of using the pseudo-...
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The volume of the unit ball in $L((\mathbb{R}^n, \| \cdot \|_2), (\mathbb{R}^m, \| \cdot \|_2))$ with the operator norm
Consider the normed space $(X_{n, m}, \| \cdot \|) = L((\mathbb{R}^n, \| \cdot \|_2), (\mathbb{R}^m, \| \cdot \|_2))$ of linear operators $\mathbb{R}^n \to \mathbb{R}^m$ endowed with the operator norm....
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Among all plane figures having a given perimeter, the circle has the largest area using the prime derivative [closed]
In relation to the question I have seen that:
Among all plane figures having a given perimeter, the circle has the largest area or, equivalently, among all plane figures having an area of assigned ...
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The measure of what sets is uniquely determined by _finite_ additivity (and translation invariance and normalisation)?
I am very familiar with measure theory but am currently wondering about how far finitary methods can take you.
Two aspects have to be differentiated: the unique determination and the calculation of ...
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A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.
Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$
Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
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Why does the volume comes out to be $\pi\frac{a^3}{8}$ instead of $\frac{3\pi a^3}{8}$?
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
I tried solving the problem as follows:
The equation ...
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
I tried solving this problem as follows:
Equation of the cylinder $x^2+(y-a)^2=...
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
I tried solving this problem as follows:
The equation ...
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Calculating Volume of Spherical Cap using triple integral in cylindrical coordinates and spherical coordinates
Given a sphere above of $xy$-plane with center $(0,0,0)$ and radius $2$ (the equation $z=\sqrt{4-x^2-y^2}$). Plane $z=\sqrt{2}$ intersect the sphere.
I want calculate volume of spherical cap (orange ...
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Volume generated by revolving $\sin x \cos x$ around x-axis
Question: find the volume generated when the region bounded by $y = \sin x \cos x, 0\le x \le \frac{\pi}{2}$, is revolved about the x-axis.
This question appeared quite tricky, and the book that ...
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Finding Volume of Revolution Given by $y = \sin x$
The question given is to find the volume of revolution generated by the graph of $y = \sin x$ on the interval $[0, \pi]$.
The way I attempted was to form the sums of cylindrical segments given by $\...
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Creating Drinking Glass using Solid of Revolution
I have to come up with two non-linear functions ($f(x)$ and $g(x)$) that will create a drinking glass when rotated 360 degrees around the y-axis.
The volume of the material of the drinking glass needs ...
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Finding the volume of a weird figure [closed]
I have a rough sketch of a problem involving a cylinder with fixed radii $r( r = 4 )$ in the picture) and that weird cross-section. The angle between the section xy plane is $\frac{\pi}{4}$. I am ...
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Question 5, section 19 of Munkres Analysis on Manifolds
Let $A$ be an open rectifiable set in $\mathbb{R}^{n-1}$. Given the point $p$ in $\mathbb{R}^n$ with $ p_n > 0 $, let $ S $ be the subset of $\mathbb{R}^n$ defined by the equation
$$ S = \{ x \mid ...
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Calculate the volume section of sphere
Look at this picture.
Given a half of sphere with radius $r=2$. Let orange plane (as picture above) is parallel with bottom plane. Orange plane is disc with radius $r=\sqrt 2$. Find the volume ...
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