All Questions
Tagged with solid-of-revolution geometry
30
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Prove that the parametric surface of revolution has continuous inverse of its image
Let $\boldsymbol{\sigma}(u,v)=\big(f(v)\cos u, f(v) \sin u, g(v)\big)$, where
$U=[(u,v):0<u<2\pi,~v\in I]$ ($I$ open interval of $\mathbb{R}$) be the parametric surface of revolution by rotating ...
- 5,340
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Spherical volumes via revolution of polynomials
In considering volumes created by revolving polynomials $y=\beta x^n$ about the y-axis, if we specify $\beta$ so that the curve includes $(0,0)$ and $(a,2a)$ and consider the volumes swept within the ...
- 2,042
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Obtaining the Surface Area of a Superegg with a Given Volume
I have been stuck trying to find an expression for the surface area of a superegg of a given volume. Specifically, the shape I'm looking at is the solid of revolution obtained by rotating a squircle (...
- 75
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Verifying formulas and process for surface area and volume of a spindle torus
While working on this geometry problem I reasoned that the surface area of the spindle torus is the surface area of the apple (outer surface) plus the surface area of the lemon (inner surface) while ...
- 133
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Is there a way to modify the solid of revolution integral to allow for solids of increasing and decreasing radius?
I am doing a project on tori as they relate to pool floaties and the volume of a normal torus can be calculated by the solids of revolution integral on a circle, Is there a way to modify the integral ...
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Volume of the solid with circular cross section
Each plane perpendicular to the x-axis intersects a certain solid in a circular cross section whose diameter lies in the xy-plane and extends from $x^2 = 4y$ to $y^2 = 4x$. The solid lies between the ...
- 370
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Sufficient conditions on the profile curve of a revolution surface to make it of class $C^k$
I will first introduce my notations then ask my questions. Thank you in advance for your answer.
Notations:
Given a surface of revolution $S_\Gamma$ of profile curve $\Gamma$ of class $C^k$ given by
$$...
- 98
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Find the Volume of a Solid Revolution around the y axis
Having trouble with this question from my OpenStax Calculus Volume 1 Homework, It is question 89 of Chapter 6 about Solid Revolution. I put my math below:
y=4-x, y=x, x=0 Find the volume when the ...
- 103
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What is the volume of the region that is within a distance r outside of the surface of an n-dimensional hypersphere of radius R?
Suppose you have an n-dimensional ball of radius R living within a d-dimensional space. Imagine the region in d-dimensional space consisting of all points that are a distance r outside the surface of ...
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On Surfaces of Revolution With Any Two Relations in $\Bbb R^2$ Such that One is the Axis (g) and the Other Revolves (f) defined by z=Rev[f(x),g(x)]:
For the last few years, I have tried a couple times to solve this problem that I came up with. Even though this may seem like a nonsensical idea, there is still a seed of wonder embedded into it.
This ...
- 12.5k
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Recurrence relation for the volume of a series of truncated cones
I'm struggling to find the recurrence relation to evaluate the volume of a solid formed by a series of truncated cones one on top of the other. The image below illustrates the problem for 2 truncated ...
- 167
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75
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Geometric meaning of surface revolution
It's known that the volume of revolution of the function $f(x)$ (assuming it's real, continuous...) is
$$V=\pi\int_a^b f(x)^2dx$$
This can be modelized as if we add together all the infinitesimal ...
- 327
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Help with a revolution solid
Suppose I have an area in the cartesian system formed by the $y$ axis and a given function $y=f(x)$. How do I evaluate the volume of the solid formed by completely revolving this area around the $y$ ...
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How to get the parametric equation of a rotated cylinder (with certain slope)
I have a basic question but I have failed in solving it. I have the equation of a cylinder which is $y^2 + z^2 = r^2$ (centered in the x-axis). The parametric equation (dependent on $L$ and $s$) is $(...
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Volume of Revolution but with higher dimensions?
I've learned in my calculus class how a function can be rotated around an axis to create a 3 dimensional shape, and the specific formulae associated with this process. What I'm wondering is whether or ...
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