Questions tagged [solid-of-revolution]
This tag is for questions regarding to "Solid of revolution", a three-dimensional object obtained by rotating a function in the plane about a line in the plane.
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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 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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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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Find the volume of solid generated by rotating this sector?
Find the volume of the solid generated by revolving the plane region bounded from upward by circle $x^2 + y^2 = 4$ and from downward by two straight lines $y = x$ and $y = -x$ around $x-axis$
So as I ...
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Generalization of why the area of a surface of revolution is not $2 \pi \int_{a}^{b} (y) dx$
Based on the posts Areas versus volumes of revolution: why does the area require approximation by a cone? and Why is surface area not simply..., approximating an ND quantity in an ND region using N-1D ...
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surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics.
I try to solve Exercise $122$ on page 40 of this pdf
Show that the surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics.
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Let $\Omega$ be the region in the first quadrant, enclosed by $y = 0$, $y = 3x$ and $y = -x^2 + 4$. Find the volume of the solid generated...
Could you help me to see if my analysis is good or wrong?
Let $\Omega$ be the region in the first quadrant, enclosed by $y = 0$, $y = 3x$ and $y = -x^2 + 4$. Find the volume of the solid generated by ...
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Volume around $y$ axis
To find the volume of the solid of revolution around $y$ bounded by
$$y=x^2,\quad y=x-2$$
and the lines $y=0$ and $y=1$, I did as follows: since the region is
Then, the volume is:
$$2\pi\cdot\left(\...
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Volume of a solid using Washer Method
The question is as follows
I'm pretty sure you have to use the washer method because the cross-section of the volume is the outer circle - inner where the inner circle: (y = 4) - (y = 1) = y = 3, but ...
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Turning a solid of revolution into a function of $x$ and $y$
I have been exploring solids of revolutions. I am trying to find different ways of expressing them to calculate their areas and volumes. For example, if a revolve the function $y=x^{\frac{1}{2}}$ ...
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Volume of Revolution but with higher dimensions?
I've learned in my calculus class how a function can be rotated around an axis to create a 3 dimensional shape, and the specific formulae associated with this process. What I'm wondering is whether or ...
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Find $f(x)$ so that volume of revolution on $[a,b]$ is $b^3-ab^2$
This is Additional Problem $26$ in Chapter $7$ of Simmons Calculus: "A solid is generated by revolving about the $x$-axis the area bounded by a curve $y=f(x)$, and the lines $x=a$ and $x=b$. Its ...
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How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted?
Why is the shell method not
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}...
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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 ...
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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 ...
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