For questions related to volume.

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3
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2answers
61 views

Volumes of Revolutions : Lord of the Rings

Question: The "Lord of the Rings" has a collection of solid gold rings for different-sizes fingers. The cross section of each ring is a segment of a circle radius $R$ as shown in the diagram below. ...
0
votes
1answer
29 views

If the volume of a container is 196 cm^3, find the dimensions of the original template.

This is a quadratics problem. The full question reads: An open container with a square base is made by cutting 4 cm square pieces out of a piece of tin. If the volume of the container is 196 cm^3, ...
1
vote
1answer
30 views

Volume by integration - Disk Method only/Non-coordinate axis

PROBLEM: Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5. (Use disk method) $$ xy = 3, y = 1, y = 4, x = 5 $$ So first I ...
0
votes
0answers
18 views

Find the new axis length of an ellipsoid that has changed its volume by delta [on hold]

I have an ellipsoid with 3 semi-axis $a$ $b$ $c$. The volume of the ellipsoid will be $V^o=4/3\pi abc$. For some reason I have to change its volume by $\Delta V$ so the new volume will be $V^n=V^o+\...
0
votes
4answers
65 views

Using Double Integral Find the volume of sphere $x^2 + y^2 + z^2= 4 $ cut by cylinder $\ x^2+y^2=2y $

Using Double Integral Find the volume of sphere $x^2 + y^2 + z^2= 4 $ cut by cylinder $\ x^2+y^2=2y $ , When i try to make integral the limits are: $\ -1<= x<=1 $ and $\ 0<=y<=2 $ ,but i ...
1
vote
0answers
14 views

Polar Coordinates Volume of Solid, Angle for integration?

I'm trying to understand how to find the angle for the integration in polar coordinate form for a solid. Here's an example of what I'm trying to solve: Find the volume of the solid bounded by the ...
2
votes
1answer
23 views

Finding the volume of Torus, Jacobian of spherical substitution.

I thought to find the volume of a Torus, like I would a sphere, where the spherical substitution was: $$x=r\cos\varphi\sin \theta , y= r\sin\varphi \sin \theta, z=r\cos \theta \\ g(r,\varphi,\theta)\...
0
votes
2answers
48 views

What is the volume of a 4d sphere

What is the volume of a 4d sphere? I've seen so many site that would have answered this question, but all of them have so many numbers that some people (including me) don't understand. So I was hoping ...
0
votes
1answer
37 views

Find an example for a bounded function $f$ where the graph of $f$ marked by ${E=(x,f(x))}$ is not of volume 0.

Find an example for a bounded function $f$ where the graph of $f$ marked by ${E=(x,f(x))}$ is not of volume 0. I tried to think about some variation of dirichlet function but the I saw that you could ...
0
votes
1answer
16 views

Volume between two surfaces with sum function

I have two functions in form: $$f(x,y)=\sum_{i}^{N}A_ig(\sqrt{(x-x_i)^2+(y-y_i)^2}) $$ where $A_i, x_i, y_i$ are different known values for both functions. $N$ is number of points. $g(r)$ is a ...
0
votes
0answers
20 views

Show that $\int_{\Bbb{R}^n}f=nV\int_{0}^{\infty}g(r)r^{n-1}dr$ for $f(x)=g(||x||)$ and $g:[0,\infty)\to [0,\infty)$ integrable

Let $||\cdot||$ be the $p$-norm with $p>1$, and let $g:[0,\infty)\to[0,\infty)$. Let $f:\Bbb{R}^n\to [0,\infty)$ be defined by $f(x)=g(||x||)$. Show that if $g$ is integrable, so is $f$, and that $...
1
vote
0answers
20 views

Cross product parallelpiped question

Hey guys can you help me with a question? So far I've got, $$P_1 = \frac{1}{{P_2}^3} (\textbf{u}\times \textbf{v})\cdot((\textbf{v}\times \textbf{w})\times(\textbf{w}\times \textbf{u}))$$
2
votes
2answers
79 views

Find the volume of the vase

I am told that the shape of a vase can be modelled by rotating the curve with equation $16x^2-(y-8)^2=32$ between $y=0$ and $y=16$ completely about the $y$-axis. I'm asked to find the volume of the ...
1
vote
1answer
30 views

Question re: derivation of formula for volume of cone

The diagram above comes from a derivation of the formula for the volume of a cone; it's one of the preliminary steps, and sadly, I'm stuck on it. What we're doing here is inscribing an "infinite" ...
0
votes
1answer
43 views

Find the Volume .

Find the volume of $(x/3)^2 + (y/3)^2 + (z/2)^2 = 36$ bounded by $x^2 + y^2 - 2y = 0$ and the plane $z=0$. was an exercise on my exams and im not sure i got it right. Sorry if im making you do the ...
1
vote
1answer
46 views

Want to confirm that I got the right answer to an “iterated integrals” question.

For positive real numbers $R$ and $r$, let $$E(R, r) = \{\frac{x_1^2 + x_2^2 + x_3^2}{R^2} + \frac{x_4^2}{r^2}\leq 1\}$$ Using an iterated integral, calculate the volume of $E(R,r)$. I am not ...
4
votes
2answers
52 views

Derivation of Shell Method

I recently saw a 'derivation' of the shell method of integration for volumes in a book that went like this: To find the element of volume contained in a shell of inner radius $r = x$ and out ...
0
votes
1answer
43 views

An open set that has no volume

Let $\mathbf Q\cap[0,1]=\{q_n\}_{n=1}^\infty$ and $A=\bigcup_{n=1}^\infty(q_n-\frac1{2^{n+2}},q_n+\frac1{2^{n+2}})$. Show that $A$ has no volume. Here "volume of a set $B$" means the Riemann-...
2
votes
1answer
40 views

Let $A=\{(x,y,z)\in \mathbb{R}^2 : x^2+y^2+z^2 \leq 1, 0\leq z \leq \frac{1}{2} \}$. Find the volume of $A$.

Let $A=\{(x,y,z)\in \mathbb{R}^2 : x^2+y^2+z^2 \leq 1, 0\leq z \leq \frac{1}{2} \}$. Find the volume of $A$. The volume I'm asked to find it's what is left of the unit semisphere minus the upper ...
0
votes
1answer
21 views

Volume between planes and a cylinder

Exercise: Calculate the volume between $ x\geq 0, y \geq 0, z\geq 0, y^2 + z^2 =1, x=2y$. I know it's a routine exercise but I fail to draw a proper graph, so my result may be wrong. Can someone ...
0
votes
0answers
26 views

is the volume of an irregular tetrahedron calculated in the same way as a regular tetrahedron

Most of the tetrahedrons I've seen have been regular and those I've seen that have been irregular I never had to do anything with. So I'm just curious.
0
votes
2answers
112 views

For $\phi(x,y)=(x,y+\psi (x))$ with $\psi:\Bbb{R}\to \Bbb{R}$ integrable, show that $\phi (B)$ is measurable for every box $B\subset \Bbb{R}^2$

Let $\psi:\Bbb{R}\to \Bbb{R}$ be integrable and define $\phi:\Bbb{R}^2\to \Bbb{R}^2$ by $\phi(x,y)=(x,y+\psi (x))$. Prove that for every box $B\subset \Bbb{R}^2$, $\phi(B)$ is measureable and $v(\phi (...
1
vote
1answer
49 views

Solid of revolution: why do I get different results for the same region?

Consider the region bounded by $y^2=x^3$, the $x$ axis and $x=4$. I want to calculate the volume of the solid of revolution of the region around $y=8$ and I can use the washer method, as the outer ...
2
votes
1answer
26 views

Volume of tetrahedron from 4 “heights”

I'm looking for a formula to find the volume of a tetrahedron from four "heights"—not the edges or vertices, but from the distance of the vertices from a common origin. (I know this is not an actual ...
1
vote
0answers
14 views

Methods to compute volume of a semi-algebraic region

I am interested in a specific question involving the computation of the volume (perhaps area is more appropriate?) of a semi-algebraic region defined by a ternary quadratic form and a ternary cubic ...
0
votes
1answer
27 views

Paradox of a unit 2 height cylinder

The surface area of a cylinder of height 2 units is $2\pi r^2 + 2\pi rh,$ which is greater than its volume $V = \pi r^2h=2\pi r^2$. Am I missing something? This is so weird!
5
votes
2answers
44 views

Volume of intersection of the $n$-ball with a hyperplane

Let $\mathcal{B}_n$ be the $n$-ball of radius $r>0$ and centre $\mathbf{x}_0$, i.e., $\mathcal{B}_n=\{\mathbf{x}\in\mathbb{R}^n\colon \|\mathbf{x}-\mathbf{x}_0\| \leq r\}$. The volume of $\...
2
votes
4answers
300 views

How did the answer key get $h=40-2r$?

A cone has radius of $20\ \rm cm$ and a height of $40\ \rm cm$. A cylinder fits inside the cone, as shown below. What must the radius of the cylinder be to give the cylinder the maximum ...
3
votes
2answers
108 views

Volume enclosed by $(x^2+y^2+z^2)^2=x$

I need to calculate the volume of solid enclosed by the surface $(x^2+y^2+z^2)^2=x$, using only spherical coordinates. My attempt: by changing coordinates to spherical: $x=r\sin\phi\cos\theta~,~y=r\...
1
vote
1answer
25 views

Complex Solids of Revolution

I know that to compute a solid of revolution of a function $f(x)$ rotated around the $y$-axis, one method we can use is the "shell" method. For example, $f(x)=1/4x^2\in [2,4]$, rotated around the $y$-...
0
votes
2answers
28 views

Volume and Average Height

How do you calculate the average number of floors of buildings across a city block when all the buildings are varying heights. Followings in a listing of what the area looks like and the number of ...
0
votes
1answer
21 views

Find volume of cube with the help of eqn of plane

The volume of cube whose two faces lie on the plane 6x-3y+2z+1=0 and 6x-3y+2z+4=0?
0
votes
1answer
25 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
0
votes
1answer
34 views

Enlargement of area and perimeter in a rotation body

Let $f: [0,1] \to \mathbb{R}$ a continuous, differentiable function with $f \ge 0$. Rotate the graph of $f$ around the x-axis. Define this rotation body in $\mathbb{R}^3$ with $A$ and the area in $\...
0
votes
0answers
6 views

On volume of arithmetic subgroups

I deal a lot with volumes of arithmetic subgroups, mainly in $SL_2(\mathbf{Z)}$. But I remain not at ease with them, making rough explicit calculations case by case instead of having a general method. ...
1
vote
4answers
48 views

A cube and a sphere have equal volume. What is the ratio of their surface areas?

The answer is supposed to be $$ \sqrt[3]{6} : \sqrt[3]{\pi} $$ Since $$ \ a^3 = \frac{4}{3} \pi r^3 $$ I have expressed it as: $$ \ a = \sqrt[3]{ \frac{4}{3} \pi r^3} $$ and, $$ \ 6 \left( \sqrt[...
2
votes
2answers
61 views

Find The Volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$

Find the volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$. $x$ and $y$ range from $0$ to $2$. $$\int_0^2 \int_0^2 y-3 \,dy\,dx $$ is correct?
1
vote
0answers
34 views

Conceptual Understanding of Rate / Volume Analysis for Balance Sheet changes

Not sure if this is the right place to ask this but I searched and didn't find this question already asked. I am having a lot of trouble conceptually understanding the formulas behind a rate / volume ...
2
votes
0answers
33 views

Spherical, polar coordinates, volume of set.

Find the volume: $$\{(x,y,z)\mid x^2+y^2 \leq (z-1)^2 \leq 4-\frac{x^2}{2} - 2y^2, z\geq 1 \}$$ I've got the intersection of the following two basically: \begin{align} 1. & & & (z-1)^2 \...
4
votes
1answer
43 views

Computing volume of concave polyhedron

I have a circular grid with points uniformly distributed throughout it. See this: Each point has some nonnegative height assigned to it (i.e. height can be 0 on up). I'm trying to accurately ...
0
votes
1answer
24 views

Finding Volume of Revolution with Multivariate Calculus

For some function $f\left(x\right)$ it is possible to rotate it along the x-axis and find the area using $$\intop_{a}^{b}\pi\left(f\left(x\right)\right)^{2}dx$$ I'm curious how to do this with ...
2
votes
1answer
30 views

Calculus II: Find the Volume (Shell-Method)

Find the volume of the shape created when rotating the region(s) bounded by $y=\sqrt{x+1}, y=0, x=0, x=1$, about the x-axis. I know this is a rudimentary question. My issue is that I tried to test ...
0
votes
2answers
29 views

I have this seemingly simple volume of a solid of revolution, but the limits and function are unknown.

How can I possibly find the numerical area of the region without knowing the function itself or the limits?
-1
votes
1answer
81 views

Volume of irregular polyhedron [closed]

Let H denotes a polyhedron with its base being a triangle PQR with PQ = QR = 1/2 unit and angle Q = 90 degree . The cover ( upper layer ) of the polyhedron is determined by the equation : z= 1/2 x - x^...
1
vote
1answer
42 views

Triple integral vs double integral to find volume of an object

Is it possible to find the volume of an object bounded by two surfaces in both of these two ways?: -a triple integral of 1 dV (I know this works) -a double integral of the top surface - bottom ...
0
votes
0answers
34 views

Find the volume(solid) , transform rectangular function to polar function

if want find volume of this problem Under $ f(x)=x^2+y^2-4 $ and inside $ x^2 + y^2=9$ in plane $z=0$ Can I use this integration in polar functions? $$\int_0^{2\pi} \int_2^{3} (r^2 - 4) r dr\,d\...
1
vote
0answers
13 views

Help finding boundries for integration

I'm trying to find the volume of the solid bounded by $y=x^3$, $y=2-x$ I have $0<=y<=1$ and $y^(1/3)<=x<=2-y$. Is there something wrong with this? I was also trying to find the volume of ...
0
votes
1answer
13 views

Unable to calculate integral for cross-section volume

I am trying to determine the volume of a solid by cross-sections, and I am having trouble determining the integral I should use to calculate it. Graph in question The base is bounded from $x = -8....
-4
votes
1answer
51 views

What is the volume of a smartie? [closed]

According to wikipedia: Smarties are oblate spheroids with a minor axis of about 5 mm (0.2 in) and a major axis of about 12 mm (0.5 in). What is the volume of one smartie?
0
votes
0answers
10 views

Half Cylinder Tank Integration Method?

I am trying to get a semi-circle cylinder tank, and I was wondering how I would set up an integral for that.∫ A semi-circle's equation, or slice, would simply be: π√(10² - x²)² Δx (set radius to 10 ...