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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What software can I use to graph a solid of revolution between 2 functions?

What software is used to graph solids of revolution between 2 functions like the ones below?
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Find the volume of y=(x^2)/8, x=2, y=0 about line x=-2 [closed]

Use disk/washer method to evaluate from 0 to 1/2
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Find the volume of the solid generated by revolving the region enclosed by $y=x^3$ and $y=x$ about $x$-axis.

I need help with this exercise. Find the volume of the solid generated by revolving the region enclosed by $y=x^3$ and $y=x$ about $x$-axis. When I graph I get that these are the enclosed regions: If ...
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Calculating Volume of Solids Rotated About y-axis and line

I have the following two "calculate the volume of the solid obtained by rotating the region" questions. Let $y = 2 - x^2$ and $y = x^4 - x^2$, and let $R$ denote the region bounded by the ...
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Volume generated by rotating axis $X =2$ and the lines $x =0$ and $y = 1$

The question I wrote the shaded area equation and add 2 to it because it has shifted two units to the right, however the answer is wrong, where did I go wrong?
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Finding the exact volume of a solid with y=5sqrtx, x=0, y=6

I just started Calculus 2 recently and while things were running smoothly, I am having trouble with volumes. At first I tried with this question the Vertical Revolution formula: 2π 0to6 (bounds) (...
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Setting up an integral that represents the volume using Washer/Shell method

Let A be the region in the first quadrant bounded by the curve $y=\cos x$, the line tangent to $y=\cos x$ when $x=\frac{\pi}{4}$ and the y-axis. a) We now rotate A with respect to the line $y=-1$. Set ...
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Find the volume of the solid of revolution formed when the region bounded by $y=\sqrt{\frac{x-1}{x^3}}$ and $y=0$ is revolved about the $x$-axis

Image description here The question is : Analytically find the volume of the solid of revolution formed when the region bounded by $y=\sqrt{\frac{x-1}{x^3}}$ and $y=0$ is revolved about the $x$-axis ...
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How to Minimize the surface area of a solid of revolution of a constant volume?

I read this post here however, I want to know whether it would be possible to minimize the surface area of a solid of revolution which is a non catenary. Catenary curve for minimum surface of ...
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Surface area of cone by integration, $\pi rh$ instead of $\pi rl$

I was trying to derive the formula for the surface area of the curved part of a cone, but ran into a problem. Let $f(x)=kx$ be any line with any angle passing through the origin, $h$ be the height (...
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Find the volume of the solid obtained by rotating the region 𝐴 in the figure about 𝑥=3.

I am stuck with this problem given that $a=3$ and $b=6.$ I was trying to cut with respect to $y$. I tried to do two integrals: from 0 to 6 and 6 to 15 because $y = 9+6$. However, I think my integral ...
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Volume of $2$ Solids of Revolution

The region bounded by $ x = y^{2}$ and $x = 2-y^{2}$, revolved around the line $x = 3$. By using the Washer method, I acquire: $$\pi \int_{-1}^{1} (y^2)^2 - (1-y^2)^2 \space dy$$ However after ...
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surface of revolution into $\Bbb R^3_{\gt 0}$

Let $\Bbb R_{\gt 0} \times \Bbb R_{\gt 0}=\Bbb R^2_{\gt 0}$ and let $\ln x \ln y=1$ be a function embedded in $\Bbb R^2_{\gt 0}.$ How would you revolve the lower branch of this function into $\Bbb R^...
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Find the volume generated by rotating the region bounded by 3 lines about the x-axis using the method of cylindrical shells

I've been solving problems on cylindrical shells and this is the formula the textbook has given me: $V=\int_a^b 2\pi xf(x)\:dx$, where $x$ is the radius of the shell. I usually just figure out the ...
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Volume of the solid with circular cross section

Each plane perpendicular to the x-axis intersects a certain solid in a circular cross section whose diameter lies in the xy-plane and extends from $x^2 = 4y$ to $y^2 = 4x$. The solid lies between the ...
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Equation of a solid of revolution

There is a way to calculate the equation of a solid of revolution, without using trigonometry, that is, a way without adding one more parameter $\theta$, because I know that it can be done with $(x,f(...
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Sufficient conditions on the profile curve of a revolution surface to make it of class $C^k$

I will first introduce my notations then ask my questions. Thank you in advance for your answer. Notations: Given a surface of revolution $S_\Gamma$ of profile curve $\Gamma$ of class $C^k$ given by $$...
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^...
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Solid of revolution about $x=9$

Find the volume of the solid of revolution of the region enclosed by the parabola $y = \sqrt{x}$, the $x$-axis and the ordinate $x = 9$, about the line $x = 9$. I assumed the professor meant ordinate $...
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2 answers
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Find the Volume of a Solid Revolution around the y axis

Having trouble with this question from my OpenStax Calculus Volume 1 Homework, It is question 89 of Chapter 6 about Solid Revolution. I put my math below: y=4-x, y=x, x=0 Find the volume when the ...
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Use the washer or shell method to find the volume of the solid obtained by rotating the region R bounded by y = x and y = x^2 about the line x = -1

I'm just confused whether I'm going about solving the above problem correctly. We're asked to set up the integrals only and simplify the integrand. Here's my process: $$= \pi \int_{0}^{1}(1+\sqrt{y})^...
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How do I use the shell method to calculate the volume of this region between two solids of revolution?

My question is how do I calculate the volume of the region between $f(x) = (2x+1)\sqrt{x^2 + x}\,$, $\,g(x) = x^2\,$, and $x = 1$ using the Shell Method? Here are the 3D visuals I graphed using ...
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Is there a quicker way to find this differential $\frac{dh}{dt}$

Just as some context, this question involved revolving the curve $y=\frac{1}{4}x^2$ around the y-axis for $y\in[0,25]$. The first part was just to find this volume which I had no problems with. Howver,...
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1 answer
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Correct way to calculate volume of solids of revolution defined by curves

When calculating the volume of a solid of revolution defined by curves, I must necessarily consider only the intersections of these curves as the region of integration (i.e all pairs $(x,y)$ ...
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Calculation of the area of the section between a plane and torus

Once I have obtained the curve of the intersection between a horizontal plane and a vertical torus (torus generated by rotation around the $x$-axis), I wish to calculate the area of such intersection, ...
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1 answer
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Calculus Disk/Washer Method for Volume

I am given the bounded functions y=ln(x), g(x)=-.5x+3, and the x-axis. The reigon R is bounded between these, and I'm tasked with finding the volume of this solid using disk/washer method when ...
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4 votes
4 answers
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Find a solid of revolution whose volume is 72π and whose surface area is 36π.

I have tried setting up multiple systems of equations using many known volumes but I always seem to come up short. My last attempt was a hollow cylinder but that leaves you with three unknowns in only ...
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Equal area parameterization of a torus?

I am trying to parameterise a surface of revolution such that each infinitesimal area element is uniform across the surface. The cross-sections of the surface are shown in the picture below. The title ...
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What is the volume of the region that is within a distance r outside of the surface of an n-dimensional hypersphere of radius R?

Suppose you have an n-dimensional ball of radius R living within a d-dimensional space. Imagine the region in d-dimensional space consisting of all points that are a distance r outside the surface of ...
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3 votes
1 answer
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solid of revolution – exercise

Here is the exercise: Find the volumes of the solids obtained if the plane regions R are rotated about (a) the x-axis and (b) the y-axis. R is the finite region bounded by $y = x$ and $x = 4y – y^2$. ...
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Volume of region in square column

Find the volume of the region in the square column $|x| + |y| \leq 1 $ and bounded by $3x+z=3, z=0$ I have attempted to find the bounds, by letting $z=3-3x, \text{ then letting } z=0, \text{ to give } ...
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2 votes
1 answer
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Compute the Volume of the Region - Confused about boundaries

Compute the volume of the region bounded by the surfaces $z=0, \ y^2=x^3 \ and \ \sqrt{x} +z=2$. I have previously computed similar questions, however there has been an integral already given, so I am ...
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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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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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2 votes
1 answer
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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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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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4 votes
3 answers
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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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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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1 vote
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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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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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1 vote
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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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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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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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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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1 vote
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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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1 vote
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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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