All Questions
Tagged with solid-of-revolution volume
149
questions
2
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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 ...
- 1,467
2
votes
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answers
57
views
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 $\...
- 1,467
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votes
1
answer
70
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Creating Drinking Glass using Solid of Revolution
I have to come up with two non-linear functions ($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 ...
0
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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 ...
- 1,159
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votes
1
answer
45
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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:
$$2\pi\cdot\left(\...
- 2,084
0
votes
1
answer
67
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How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted?
Why is the shell method not
$$\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)}...
user1291046
5
votes
1
answer
66
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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 ...
- 2,761
4
votes
2
answers
139
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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 ...
- 751
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votes
1
answer
27
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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 ...
0
votes
1
answer
47
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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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1
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790
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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 ...
1
vote
1
answer
54
views
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 ...
- 1,467
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1
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61
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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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3
votes
1
answer
110
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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 ...
- 1,043
0
votes
2
answers
47
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Volume when the region bounded by $y= -\frac{1}{4}x^2 + x, y= -\frac{1}{8}x^2 + x,$ and the $x-$axis about the $y-$axis.
Find the volume of the solid of revolution obtained by revolving the first quadrant plane bounded by $y= -\frac{1}{4}x^2 + x, y= -\frac{1}{8}x^2 + x,$ and the $x-$axis about the $y-$axis.
Attempt:
...
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