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.
303
questions
0
votes
0answers
25 views
Graphing rational expressions of trig/exp by hand?
In my math class, we are not allowed to use calculators. Thus, when I see problems with an exponent/log or trig either in top or bottom and I have to graph them to solve the problem, I'm not sure ...
0
votes
1answer
33 views
Let $f(x)\geq 0$ be continuous on the interval $[a, b]$. Consider the solid of revolution by rotating the graph of $f(x)$ around the $x$-axis. Prove
$1.$ Let $f(x)\geq 0$ be continuous on the interval $[a, b]$. Consider the solid of revolution by rotating the graph of $f(x)$ around the $x$-axis. Prove that its volume is
$$V=\pi \int_a^b [f(x)]^2 \,...
0
votes
1answer
42 views
GRE subject:volume created by revolving area about line
find volume created by revolving the region between $x=y^2$ and $y=x^2$ about the line $x=-1$.
My attempt is since to solve for $x$ first correct? so we have $x=y^2, x = \sqrt{y}$ so we. integrate ...
2
votes
1answer
39 views
Sphere parameterization from the general parameterization of a surface of revolution
I'm trying to derive the parameterization of a sphere from the general parametric equations for a surface of revolution. In particular, I read on wikipedia, that in general, to parameterize a rotating ...
0
votes
1answer
26 views
extrema of volume of solid of revolutions
Based on the definition of the volume of a solid of revolution, i wanted to apply Euler equation and find the extrema as follows:
$$
v = \pi\int^{a}_{b}{y^2dx}
$$
using euler equation:
$$
\frac{\...
0
votes
1answer
22 views
How do you express a surface of revolution as the graph of a function?
Consider a real function $f$ with domain non-negative real numbers.
Let $y = f(x)$, and consider the surface traced out
by rotating the graph of $f$ about the $y$-axis.
This surface is the graph of ...
4
votes
3answers
98 views
What is the correct formula for the washer method?
Almost everywhere I look the formula is:
$$ \pi \int_b^a {\left(f(x)^2 - g(x)^2\right) dx} $$
where f(x) is the big function and g(x) is the smaller function.
Though I've run into problems while ...
0
votes
1answer
18 views
Washer method, axis intersecting with region itself
I know the basic idea of disc / washer / cylindrical shell methods, to find the volume of solids generated : be it a single function, or the bounded region between two functions.
The trouble I am not ...
1
vote
1answer
32 views
Solid of revolution problem - clepsydra (water clock)
I need help to understand this problem:
A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The "clock" is calibrated for ...
0
votes
0answers
32 views
Finding the volume of revolution obtained by rotation of a region about the line $2x-y=20$
I have the following question before me: Find the volume of the area bounded by the curves $y=x^3$, $x+y=10$ and $x=0$ about the line $2x-y=20$. I started off by drawing the given area and line of ...
0
votes
1answer
24 views
Finding the f(y) in solving volume of solid revolution
Good day.
I am trying to solve this question:
"Find the volume of the solid obtained by revolving the indicated region about the given line. The region is bounded by the curves $y=x\sqrt{2-x}$ , $...
1
vote
0answers
47 views
Integral of volume of a solid of revolution
Hello! I was practicing for my upcoming math test, but I wasn't sure about this problem. I think I did it right, but I just wanted to check to be safe. I got answer choice B. Would anyone know if that ...
0
votes
1answer
46 views
Volume generated by shell method of y = 4
$$\mathrm{x = 4y-y^2, x = 0, y = 4…}$$ Volume by shell method.
I am just confused with everything. I cannot figure out, the radius, and height. Any help would be appreciated thanks.
The radius I took ...
-2
votes
1answer
22 views
How to find the volume by using shell method [closed]
I would like to find the volume of $\frac {x^2} 9+y^2=1$ rotated around the $x$ axis, but using the shell method. How can I do this?
Letting the thickness be $dy$ and the problem I encountered is ...
-1
votes
1answer
30 views
Help with volume of solids of revolution [duplicate]
I have a region $R$ defined by $y=x^2$, $y=2+x$ and $x=0$.
How would be the integrals (no need to develop them) of the solid obtained by the revolution of $R$:
$a)$ around $x$-axis integrating in ...
0
votes
1answer
31 views
Volume of solid of integration - trying to spot a silly error!
Here is a question:
Find the volume of the solid formed by rotating the region enclosed by the graph of the function and the $x$ axis through $2\pi$ radians about the $x$ axis in the given integral:
y ...
1
vote
1answer
122 views
Finding the volume of a solid of revolution when the function in question is an inverse trigonometric one
Here are two questions:
Find the volume of the solid of revolution, generated by rotating the region bounded by the graph of $y = \arcsin x$ and the lines $x = 1$ and $y = 0$ through 2$\pi$ radians ...
1
vote
0answers
34 views
Volume of a solid generated by rotating a curve around a given line
The curve is $y=\frac{1}{1+x^2}$, on the interval [1, 4] and the line is x = -1.
The integral I have set up using the cylindrical shell method is
$$ 2\pi \int_1^4 \frac{x+1}{1+x^2}\, dx $$
but I'm not ...
2
votes
1answer
101 views
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 ...
1
vote
1answer
65 views
How would I find the volume of a paraboloid, using volumes of revolution, with only the equation of the paraboloid?
I have been given an equation describing the surface of an open paraboloid, z = a^2 - x^2 - y^2. In this case, z is the 'vertical' axis, and a is some constant. z is also greater than or equal to 0.
I ...
1
vote
1answer
64 views
Using shell method to find volume of solid about x axis
Find volume of solid obtained by rotating the regin formed by $\displaystyle y=\frac{1}{x}$ and $x$ axis and line $x=3$ to $x=4$ about $x$ axis using Shell method is
What i try::
I have tried using ...
1
vote
0answers
18 views
Volume generated by revolving region of $y=-1,y=\ln x,x=y^2+1$ about $y=1$
I am trying to find the volume of solid generated by revolving region of $y=-1,y=\ln x,x=y^2+1$ about $y=1$ using shell method.
here is the picture.
I need help in setting up the integral?
I actually ...
2
votes
1answer
36 views
Integration application to volume
I'm trying to find out the volume of:
Area enclosed by $y=\arctan x, \ y=0, \ x=1$ rotated about y-axis.
I tried to use both the disc and cylindrical method, and I can get it with the cylindrical ...
0
votes
1answer
31 views
Is there problems where the shell method cannot be applied?
The question is to find the volume of the solid formed by the revolution of the area $A$ around the $x$-axis. And the area $A$ is bounded by the curve $y=x^{\frac{1}{4}}$ and $y=x$ and $x=0$ and $x=1$....
6
votes
1answer
59 views
Disc vs Shell Method, getting different answers AP calc
Can someone please check my work.
$R$ is the region in the first quadrant bounded by $y=1/x$, $y=1$ and $x=e$
Find the volume of the solid generated when $R$ is revolved about the line $y=1$
Disk:
$$V=...
3
votes
1answer
101 views
Can someone check my solution for AP Calc Volume of a Solid question
Let $R$ be the region in the first quadrant bounded by $y=1/x$, the horizontal line, $y=1$, and the vertical line $x=e$.
Region $R$ is the base of a solid. For this solid, each cross section ...
3
votes
0answers
37 views
Volume of revolution about $x=3$
Find the volume generated by revolving the region shown below about $x=3$ using Washer method.
We have outer radius as a function of $y$ as:
$$R(y)=3$$
Inner radius as a function of $y$ as:
$$r(y)=3-\...
1
vote
2answers
40 views
Volume of solid of revolution by revolving the region $y=x^2$,$x=0$,$y=9$
Find the volume generated by revolving the region shown below:
I have done as follows:
Revolve the rectangle formed by the vertices $(0,0),(3,0),(3,9),(0,9)$ about $X$ axis.
The required volume ...
0
votes
1answer
56 views
Showing that a solid of revolution is Lebesgue measurable
Let $E \subset \mathbb{R}^2$ be Lebesgue measurable in $\mathbb{R}^2$, bounded and in the right half-plane: $(r,z) \in E \Rightarrow r >0.$ $R(E)$ is rotating around the $z$-axis.
$$R(E):=\{(x,y,z)...
0
votes
1answer
64 views
Struggle with obtaining same answer for both Disk-Washer Method and Shell Method on same Problem
The problem is straightforward with the Disk-Washer method. To help me fully understand these rotation problems I like to work the problems using both Disk-Washer and Shell (some problems become too ...
1
vote
0answers
39 views
Find the volume by revolution of the integral Mathematics
If we have the surface $S$ delimited $Y$ and the following functions, what is the volume found by the revolution of the surface $S$ around the $Y$ axis.
$f(x)= (x- 5)^2$
$g(x)= x^2+ x+ 3$
We have ...
0
votes
1answer
64 views
What's wrong with my Surface Area of a solid of revolution formula?
When I learnt about the derivation for the formula $$V=\pi\int_{x_1}^{x_2} y^2~dx$$
where $V$ is volume of the solid generated when $y=f(x)$ is rotated about the $x$ axis by $2\pi$ radians between $...
3
votes
2answers
72 views
Use washer and shell method to find $k$
Let $R$ denote the first quadrant region bounded by $y=k-kx^2$, and $y=kx^2$. $k> 0, k \in \mathbb{R}$. If $V_1$ is the volume obtained by rotating $R$ about the line $y=0$, and $V_2$ is the volume ...
0
votes
1answer
24 views
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 ...
0
votes
1answer
25 views
Volume related to rotating over y=-1
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the following curves about the line y=−1.
$y=x^2$ and $x=y^2$
Visual of ^^
What I got so ...
2
votes
0answers
34 views
finding the volume of the solid of revolution for $\frac{x^n}{n!}$
I'm trying to find the volume of the solid of revolution formed by revolving the curve whose equation is given around the X-axis between the points indicated. I would like your feedback on my solution....
0
votes
0answers
270 views
Find the moment of inertia of the solid of revolution of a cardioid
We are given a cardioid $r=a(1+\cos \theta)$ of constant density $\rho$ and it is rotated about the initial line ie. $x$-axis and we are supposed to find the moment of inertia of the solid so obtained ...
2
votes
2answers
131 views
Volume of solid of revolution about a line other than the axis - using Cylindrical Shells method
I am struggling with a concept of finding the radius in the shell method of finding the volume of the solid when revolved around a line other than the axis. Maybe the issue here is that I am looking ...
0
votes
1answer
16 views
parabolic volume integration mistake
While dealing with volume integral of paraboloid bowl, say the volume within the surface $x^2 + y^2 = r^2 $ and $z = r^2$, where $ 0\le r\le \sqrt{3} $. One classic way seems to firstly have ...
0
votes
1answer
56 views
Rotation about the y-axis and volume between two-curves.
I was doing this practice question that if given two functions of y=1/4x and y=x^2, find the volume of the solid around the y-axis.
I am just really confused on what to do - could someone please guide ...
4
votes
2answers
120 views
Using shell method, find the volume of the solid generated when $y=3-x^2$ is revolved around x-axis from $y=0$ to $y=3$
Question
Via the Shell Method, what is the volume of the solid of revolution formed by revolving a portion of $y=3-x^2$ about the x-axis bounded by $y=0$ and $y=3$.
Context
The answer from Wolfram ...
0
votes
0answers
30 views
Volume of Solids Generalized
I want to make sure I'm understanding this correctly.
If a continuous, nonnegative function $f(x)$ on $[a,b]$ was revolved about some axis, where it grows by $x$ in $\Delta x$ intervals, then the rate ...
1
vote
2answers
27 views
Help! perplexing problem on volume revolution of region about a line
how do we find the volume when the region bounded by $y = x^{\frac 1 2}$ and $y = \frac x 2$ is revolved about the line $y = 1$? I understand how to solve for the regions above and below $y = 1$, ...
1
vote
0answers
25 views
Limits for the parameters of the parametric equation of a surface of revolution
According to Lipschutz's Differential Geometry book,
A surface of revolution $S$ is obtained by revolving a plane curve $C$ (the profile curve) about a line $L$ (the axis of $S$) in its plane. If $x_{...
0
votes
1answer
68 views
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$$
This can be modelized as if we add together all the infinitesimal ...
0
votes
1answer
27 views
Help with a revolution solid
Suppose I have an area in the cartesian system formed by the $y$ axis and a given function $y=f(x)$. How do I evaluate the volume of the solid formed by completely revolving this area around the $y$ ...
1
vote
2answers
49 views
Shell method to compute volume
I'm currently learning about the shell method to compute the volume of a solid of revolution. I am working on the following problem:
Find the volume of the solid obtained by rotating the region by $y =...
2
votes
2answers
134 views
How do i solve this integration question using the washer and shell method?
What is the volume of a solid enclosed by $y = (x-1)^2$ and $y = 4$ revolved around $x = - 3$?
I tried the washer method and the shell method and got different answers each time and I'm really ...
1
vote
1answer
51 views
Find the volume of the solid by rotating the region bound by $y = 1 - x^2$, $y = 0$, and $x = 0$ about the line $y = 2$
I am not really sure if I am on the right track right now. But I am supposed to find the volume of the solid.
My Attempt (Volumes of Revolution):
So I am given the following equations:
$$\begin{align*}...
0
votes
0answers
860 views
Revolving an area between two curves around $y = -3$ without knowing the bottom function
The function $f$ is defined by
$$f(x)=3(1+x)^{0.5}\cos\left(\frac{\pi x}{6}\right)$$
for $0\leq x\leq3$. The function $g$ is continuous and decreasing for $0\leq x\le3$ with $g(3)=0$.
The figure ...
