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.
408
questions
4
votes
2
answers
139
views
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
2
votes
0
answers
70
views
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 ...
0
votes
1
answer
176
views
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 ...
- 11
1
vote
1
answer
49
views
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 ...
- 21
1
vote
2
answers
152
views
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
0
votes
1
answer
27
views
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
views
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 ...
- 3
0
votes
1
answer
790
views
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
0
votes
1
answer
61
views
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\...
- 3
0
votes
1
answer
52
views
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,\...
- 872
3
votes
3
answers
113
views
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 "...
- 87
2
votes
2
answers
57
views
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 ...
- 7,778
6
votes
4
answers
270
views
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 ...
- 1,043
3
votes
1
answer
110
views
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