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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volume of solid obtained by rotating the curve about $x=2$ line

Finding volume of solid obtained by rotating the regin enclosed by $y=x^3, y=0, x=1$ about $x=2$ line is What i try:: Volume of solid $$V=\pi\int^{8}_{0}\bigg[(2-x)^2-1\bigg]dy$$ $$V=\pi\int^{8}_{0}\...
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2answers
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Different Volume Using Shell & Disk Methods

I've been trying to find my mistake for a while, but I can't seem to find one. Maybe I'm missing a hole that is present? I tried the volume of the region bounded between $x=1$, $x=3$ and a linear ...
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0answers
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Finding a volume generated by a parabola

Below is a problem I did. The book gets $\frac{16\pi}{15}$.This number seems to large to me. I am hoping that somebody can confirm that I got it right or tell me where I went wrong. Problem: Find the ...
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1answer
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Finding a volume generated by a parabola two ways

Problem: Find the volume generated when the region bounded by the given curves and lines is revolved about the x-axis using the disk method. Then find it using the cylindrical shell method and verify ...
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1answer
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Finding a volume using the method of cylindrical shells which is generated by a parabola

This problem is from the 7th edition of the book "Calculus and Analytic Geometry" by George Thomas and Ross Finney. It is problem number 3 of section 5.4 Problem: Find the volume generated ...
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1answer
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Volume of Solid of Revolution with missing variable

The region bounded by the parabola $y=L·x·(2-x)$ and by the lines $y=0$ and $x=1$ is shown below. https://i.stack.imgur.com/BP0L2.jpg When this region is revolved about the line $y=0$ a solid of ...
2
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2answers
39 views

Volume of a circle $x^2 +y^2 \leq 1$ which is revolving around a line $x+y=2$.

I want to compute the volume of a circle $x^2 +y^2 \leq 1$ which is revolving around a line $x+y=2$. Usually I solved problems about solids revolving around axis and non axis horizontal and vertical ...
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Help with minimal surface of revolution: symmetric case

I posted this question on the physics stack exchange, but I have got no answers in a few days. I was hoping that the mathematicians would be more helpful! I am trying to solve exercise 7 from ...
2
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0answers
29 views

Volume of Revolution but with higher dimensions?

I've learned in my calculus class how a function can be rotated around an axis to create a 3 dimensional shape, and the specific formulae associated with this process. What I'm wondering is whether or ...
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2answers
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Why use different intuitions for volume and surface of revolution.

Suppose $y=f(x)$ is a continuous curve on $[a,b]$.Suppose we are to find the volume of revolution of the solid generated by the area under $f(x)$ and bounded by $x$-axis and the ordinates $x=a$ and $x=...
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1answer
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Why does the integral for surface area require an expression for arc length? (Solid of Revo.)

I'm a lowly Calculus II student here and I noticed something interesting/confusing. The integral formula for the volume of a solid of revolution (SoR) performed with the disk method has the form $$ V=\...
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Formalize an observation about centroid of solid of revolution

Let's suppose to have a figure $\Bbb F$ on the plane –to simplify– $(y, z)$; and to revolve it of an angle $\alpha$ about $(z)$-axis. Sometimes it happens that it's not needed to evaluate explicitely ...
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1answer
29 views

Solid defined in 3 dimensions [closed]

Which shape is defined as follows: $A=\{ (x,y,z) \in \Bbb{R}^3 : 0\le{y}\le{1}, 0\le{z}\le{1}, z\le{x}\le{z+1}\} $ I struggle picturing these things.
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2answers
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Intersection of a line with a specific surface of revolution

As part of an art project, I'm trying to write a program to raytrace eggs, as defined by the model described in this paper. As part of writing this program, I am trying to calculate the intersection ...
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2answers
62 views

Centroid of volume of revolution

Consider a solid generated by the curve $y^2 =ax^2+2bx+c$,rotated about the $x$-axis, and two plane surfaces at right angles to the latter, distance $h$ apart, and with areas $A$ and $B$. To prove ...
2
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1answer
36 views

Find the length of the barrel and the greatest and least values of the diameter.

The shape of a barrel is made by rotating the segment of the curve $y=40-0.002x^2$ between $x=50$ and $x=-50$, around the x-axis. The units are in centimeters. What is the length of the barrel and ...
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0answers
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What is the 2D shape that gives the maximum finite volume of a solid of revolution?

What is the 2D shape that gives the maximum finite volume of a solid of revolution? The following image (inspired by the one at Weisstein, Eric W. "Pappus's Centroid Theorem." From MathWorld-...
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2answers
38 views

Finding the height of a Pyramid where the sides are given by an equation

Problem: The vertex of a pyramid lies at the origin, and the base is perpendicular to the x-axis at $x = 4$. The cross sections of the pyramid perpendicular to the x-axis are squares whose diagonals ...
4
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2answers
33 views

Find volume of the solid generated by revolving

Find volume of the solid generated by revolving the region bounded by the parabola $ 𝑥=𝑦^2+1 ,𝑦=0 $ and the line 𝑥=3 about the line 𝑥=3 Using Disk method,I found the answer to be 9.4
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1answer
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Find the function $f(x)$ knowing the volumen of the solid of revolution for $a \gt 1$

Find the function $f(x)$ knowing that the volumen of the solid of revolution of $y=f(x)$ around the x axis, for $a \gt 1$ is $V=a^2 -a$ , $f(x)$ continuos and $ f(x) \gt 0$ Basically what I tried to ...
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0answers
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How do you find the function when the volume of solid revolution is given?

Here is the question: For the area between $x=g(y)$, $y-$axis, and $y=a$ and $y=2a$ lines, the volume is known to be $a^2$ when the area is rotated around the $y$-axis. Similarly, for the are between $...
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1answer
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Why for determining the the surface area,the function should be smooth over the interval?

Let $f(x)$ be a non-negative smooth function over the interval $[a,b]$ . Then, the surface area of the surface of revolution formed by revolving the graph of $f(x)$ around the $x$-axis is given ...
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Finding the constant that results in the volume of the solid

Good afternoon, I came across the following problem and I couldn't find a way to solve it "Given the following pieces of information: $a>0$ $R=${$(x,y) \in \mathbb{R}^2|x \geq 0,a \leq y\leq −...
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2answers
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Calculate the volume of the solid generated by two regions [closed]

How can I calculate the volume of the solid generated by the S1 and S2 regions by rotating around the Ox axis and around the axis Oy? S1 : $0 ≤ x ≤ 2$ $0 ≤ y ≤ 2x − x^2$ S2: $0 ≤ x ≤ π$ $0 ≤ y ≤ \sin^...
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1answer
35 views

Proving the surface area and volume of revolution formula

Title Is there any way to prove the formula or surface area and volume of revolution of a function? The derivation of the formula I have found online does not really seem like a proof, but more like ...
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1answer
46 views

Find the volume of the shape rotating $x$-axis.

Write and sketch the integral of intersection of $$y=\sqrt{x}, \ y=\sqrt{8-x}, \ y=1$$ Then find the volume of the shape rotating around the $x$-axis. I think there is something wrong while ...
2
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1answer
29 views

Volume of a solid of revolution with change of variable

I want to calculate the volume of the solid of revolution around the x-axis of this figure $x = (1-t^2)/(t^4+4)$ $y = (t+1)*(1-t^2)/(t^4+1)$ for t between -1 and 1. In the figure below the plot is ...
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1answer
52 views

How can a function have a finite area but infinite volume when revolved around an axis?

I have a function $f(x) = \frac{1}{\sqrt{x}}$ on $[0, \pi]$ and want to find the area of the function, then the volume of the function were it revolved around the x-axis. For the area: $$\begin{align} ...
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0answers
21 views

Volume of the solid obtained by revolving

To see the question please click here I found the function is second root of $(x-1)$. The question I can't find the answer is that: Find the volume of the solid obtained by revolving the region $...
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1answer
30 views

Find the volume of a specific region of a solid of revolution about the y-axis

I have a given polynomial, f(x)=c1*x^7+c2*x^6+c3*x^5+c4*x^4+c5*x^3+c6*x^2+c7*x+c8 rotated about the y axis, which results in a nice surface: What I would like to be able to do is evaluate the volume ...
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2answers
31 views

Stuck on volume of a solid of revolution [closed]

The problem I've been working on is where the curves are y=sqrt(x), y=1/x, and x=5, which are rotated around the y-axis. I am able to do this problem when rotating around the x-axis, but I have no ...
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1answer
35 views

Distance between two points in a solid of revolution

Let $f: I \to (0,\infty)$ be a smooth and positive function. Let $$\Sigma_f = \{(x,f(x)\cos\theta,f(x)\sin\theta) : x \in I, \theta \in \Bbb{R}\}$$ be a solid of revolution generated by rotating ...
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1answer
34 views

Finding volume of solid in one quadrant - divide total volume by 4? 8? 2?

I want to find the volume of the solid produced by revolving the region enclosed by $y=4x$ and $y=x^3$ in the first quadrant. The wording about the first quadrant confuses me but here's my work so far:...
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2answers
16 views

Volume of Solids of Revolution with Hyperbola

The area bounded above by the line $y = 3$, below by the line $y = 0$, on the left by the y-axis and on the right by an arc of the hyperbola $9x^2 - 16y^2 = 144$ is rotated around the x-axis. Find the ...
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1answer
29 views

find the volume using the method of disks or washers via an integral

The volume of the solid obtained by rotating the region enclosed by $y=\frac{1}{x^2} , y=0, x=3, x=8$ about the line $y=-1$. How do I find the volume? I need help. I tried $\pi\int_8^3(\frac{...
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1answer
12 views

If the region $D$ is revolved about the $z$-axis in $ℝ^3$, then the volume of the resulting solid is

Consider the region $D$ in the $yz$ plane bounded by the line $y=\frac{1}{2}$ and the curve $y^2+z^2=1$, where $y\geq 0$. If the region $D$ is revolved about the $z$-axis in $ℝ^3$, then the volume of ...
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1answer
38 views

Incorrect answer in integral of volume.

Problem: Let $S$ the solid limited by the surfaces \begin{align} x= \sqrt{y^2 + z^2 }, \quad x=\sqrt{\frac{y^2 + z^2}{3}}, \quad x=\sqrt{a} \end{align} the value of parameter $a$ for the following ...
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2answers
48 views

Volume of a Solid of Revolution?

Fairly quick question, but I'm a bit confused on whether or not the work I did was in any way right. So the question went something like this: "Find the volume if the region enclosing $y=x^3, x=0,$ ...
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1answer
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why isn't the surface area of revolution equal to the the sum of the perimeters of the circles that makes it?

let $w$ be the width of a rectangle and $l$ be the length. From the definition of area it sounds logical to add up $w$ lengths of a rectangle to find how much two dimensional space it is occupying. ...
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1answer
27 views

Find the equation of the surface of revolution.

Find the the equation of the surface revolution obtained by rotating the following curve around y axis. $$ x^{3/2}+y^{3/2}=1 $$ I tried to form the f(y) from the equation above and form the ...
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1answer
25 views

How do you find the volume obtained by rotating $0≤y≤8\sqrt{x - x^2}$ about the $y$-axis within $0≤x≤π$?

I can't seem to obtain a simple way expression of $x$ in terms of $y$, and the integration is very complex otherwise.
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1answer
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Integration with volume finding $t$ [closed]

A solid formed by rotating the curve $$y = x^2 +4 $$ between $x =1$ and $x =t$, $t>1$, through $360^{\circ}$ about $y$-axis. Find the value of $t$ for the volume of solid formed is $40\pi$.
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1answer
54 views

Volume of a weird shape

Here it's a question actually on integration but I simply don't know how to calculate the volume of the cone enclosed between the dotted planes. It might be very trivial but I could get up to the only ...
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1answer
36 views

Show surface of revolution is an orientable surface

If I have a curve parametrized in the form $(f(v),0,g(v)),f>0, v\in(a,b)$, rotating this curve about the $z$-axis. I get a surface of revolution $S$ and I know it is a regular surface as it can be ...
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0answers
29 views

Calculus 2: Find the volume of the solid generated by revolving the region under the curve y = 9 / x

I solved the following problem but my answer does not match the textbook answer. ...
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1answer
29 views

Is this the volume of a solid of revolution of a sector about a point in space along phi

Consider this sector S what's area is the following: $$ A = \frac{1}{2}r^2\theta $$ Where theta is in radians. I would like to create a solid out of this sector by rotating it about $\phi$ with the ...
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1answer
27 views

Integrating a function which cannot be isolated

I want to revolve the region bounded by $y=1$, $x^2=y^2+y$ about the $x$ axis. For this, I need to isolate $y$ in terms of $x$. Then, $V = \pi\int^{\sqrt2}_0 (1-y^2)dx$. Is it possible to set $y$ and ...
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0answers
17 views

Find the volume of the solid generated by revolving the region bounded by $y=\sin(x)$ and $y=0$ about $y=-x$

I need help computing the volume of this problem without Pappus Theorem. I know how to proceed when I revolve the curve about a vertical line or horizontal line, but I don't know how to solve it when ...
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1answer
31 views

Volume of a solid by revolution (cylindrical shells)

I frequent Stack Overflow, but I am new here. I was given this problem: Use the method of cylindrical shells to find the volume $V$ generated by rotating the region bounded by the given curves about ...
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0answers
15 views

Express y=0 in terms of x to find volumes of revolution?

I'm in calc 2 and have to solve this problem: Consider the region bounded by y=x, y=0, x=1, and x=3. Solve for V, the volume of revolution of this region around: a) y = -1 using the disk and washer ...

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