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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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 ...
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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 \,...
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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 ...
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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 ...
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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{\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $y=x\sqrt{2-x}$ , $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $x=1$....
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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=...
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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 ...
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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-\...
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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 ...
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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)...
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1answer
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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 ...
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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 ...
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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 $...
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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 ...
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1answer
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 $y = 1$, ...
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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_{...
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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 ...
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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 $y$ ...
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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 =...
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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 ...
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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*}...
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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 ...

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