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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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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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:
$$2\pi\cdot\left(\...
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Volume of a solid using Washer Method
The question is as follows
I'm pretty sure you have to use the washer method because the cross-section of the volume is the outer circle - inner where the inner circle: (y = 4) - (y = 1) = y = 3, but ...
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Find $f(x)$ so that volume of revolution on $[a,b]$ is $b^3-ab^2$
This is Additional Problem $26$ in Chapter $7$ of Simmons Calculus: "A solid is generated by revolving about the $x$-axis the area bounded by a curve $y=f(x)$, and the lines $x=a$ and $x=b$. Its ...
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How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted?
Why is the shell method not
$$\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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Find surface which generated by revolving a line in $\mathbb{R}^3$
Problem :
Let $l$ be a line which passes two points : $(1,0,0), (1,1,1)$.
And $S $ be a surface which generated by revolving line $l$ around $z$-axis.
Find a volume enclosed by surface $S$ and two ...
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Finding the equation of two lines lying on the surface of a hyperbloid
Trying to solve this question:
Surface S is obtained by revolving the line $x^2-z^2=4$ around the "z" axis.
Write the equation for S.
Show that exactly two lines pass through M=(2,0,0) ...
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Confusion about the bounds for solids of revolution
Use the reduction formula for the integral:
$$\displaystyle I_{a,n}= \int_{0}^{1} u^ne^{au} du, a\in\mathbb{R} ,a\neq0,n=0,1,2,\dots $$
to calculate the volume of the solid of revolution obtained by ...
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Evaluating $\int_1^e{\sqrt{\ln x}}dx$ by finding volume
$$\int_1^e\sqrt{\ln x}\;\mathrm{d}x$$
WolframAlpha provides an answer to the integral in terms of the imaginary error function. However, I was wondering why the method I employed did not work:
I can ...
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Surface (superior and lateral) and volume of an ungula
Context
Definition: An ungula is the solid obtained by cutting a cone with a plane and keeping the part between the base of the cone and the plane
I couldn't find the formulas to obtain the upper ...
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Volume of the solid using cylindrical shell method
The region $R$ is bounded by the $x$-axis, the vertical lines $x = \frac 12$ and $x = a$ for some $a > 1$; and the graph of $y = \frac 1\pi(e^{x^2-x})$. Find, in
terms of $a$, the volume of the ...
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Volume of Rotation Between Two Solids
Suppose $R$ is the region in the first quadrant bounded by $y = 2+x$, $y= x^2$, and $x=0$. I was supposed to find
(a) the volume of the solid generated by revolving around the $y$-axis and
(b) the ...
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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 (...
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How can I estimate the volume of a solid object, knowing only it's longitudinal corss-sectional area?
Let's say the shape is too complex to split it into simpler parts and solve it analytically.
I can obtain it's longitudinal cross-sectional area by loading the image into an image editor, scaling it ...
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3D Volumes of Revolution
So I was wondering how I could graph 3D Volumes of Revolutions on Graphing softwares for my Investigation, but I am not sure how to do it, I have seen some youtube and geogebra links but how do I do ...
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Volume of revolution of solid formed by $y=x^2$ and $y=2x$ about $y=-1$
I'm trying to find the volume of the solid obtained by rotating the region between the curves $y=2x$ and $y=x^2$ around the line y=-1 .
This is what the graph looks like
I'm mainly struggling due to ...
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Is An Infinitely Thin Cylindrical Shell a Rectangle?
Yesterday I finished reading the method for finding the volume of a solid of revolution using cylindrical shells, the textbook I use of course gave a rigorous proof on why it works, however, it also ...
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Why can't we use discs with 'slanted edges' when calculating the volume of a solid of revolution?
For example, to find the area of a hemisphere of radius $R$, I think of stacking discs with radii $r=Rcos(\theta)$ and side length $Rd\theta$, so the area of each disc is $dA=2\pi R^2cos(\theta)d\...
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Local isometry between half a disk and the cone of revolution $3(x^2+y^2)=z^2$
This is an exercise from my Differential Geometry course:
Define the function $\Phi:\left]0,2\right[\times \left]-\pi/2,\pi/2\right[\longrightarrow \mathbb{R}^2$, $\Phi(\rho,\theta)=(\rho\cos \theta,\...
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Attempting to compute *surface* of solid of revolution
I saw that in order to compute the volume of a surface of revolution, we can use $\int_a^b\pi f^2\left(x\right)dx$, where $f$ is the curve to be rotated. This seemed really intuitive: for each "...
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On bodies of revolution for $y =(1-x^q)^p$
This question is posted in response to a recent one seeking the volume of $y =(a^{2/3}-x^{2/3})^{3/2}$ rotated about the x-axis. I wondered why people don't seek a more general solution when posed ...
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Why this solids also lives below z axis?
The base of a certain solid is the circle $x^2 + y^2 = a^2$.
Each plane perpendicular to the x-axis intersects the solid in a square cross-section with one side in the base of the solid.
Find its ...
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Volume of tent with a circular base and stretched over a semicircular rod
A tent consists of canvas stretched from a circular base
of radius "a" to a vertical semicircular rod fastened to the
base at the ends of a diameter. Find the volume of this
tent.
I was ...
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