Questions tagged [volume]

For questions related to volume, the amount of space that a substance or object occupies.

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Calculate the volume of rotation of $C: x = t^2, y = 2t$ and the line $y = x$

I am working on a problem (Pearson Core Pure Mathematics, Book 2, Chapter 4 Mixed Questions, Challenge) and cannot understand where I am going wrong. Below I will post 2 separate methods of mine, ...
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2 answers
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Why is the volume of the solid of revolution between two graphs $\int_{0}^{b}\pi[f^2(x)-g^2(x)]dx$ instead of $\int_{0}^{b}\pi[f(x)-g(x)]^2dx$?

Let $f$ and $g$ be two non-negative functions which are integrable over the interval $[0,b]$ and satisfy $f \geq g$ on $[0,b]$. I am told that the volume of the solid of revolution created by rotating ...
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1 answer
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Finding the volume of liquid in a spherical tank

Suppose there is a spherical tank partially filled with a liquid so that the liquid has a maximum depth of h. I want to find the equation for the volume of liquid in a tank in terms of depth h (and R ...
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Why is the greedy algorithm philosophy to finding the maximum right cylinder inside a rectangular box not correct?

The largest solid right cylinder possible is placed in a rectangular box measuring 8 by 10 by 12 inches. If the box is then closed, approximately what percentage of the box's volume is occupied by the ...
3 votes
2 answers
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Is volume of semialgebraic sets definable over the language of ordered fields?

Let $ \mathcal{L}=\{0,1,+,\cdot,\le\} $ be the language of ordered fields and consider the theory of $\mathbb{R} $ in this language (i.e., the theory of real closed fields). Suppose $ \varphi(x_1,\...
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1 answer
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Volume of region bounded (with 3 bounds?)

I need someone to walk me through to the right answer as I don't know how to reach it. Question is as follows: Find the volume of the region bounded by $ y 1 = x 3 + x , y 2 = 0 $, and $ x = 2 ...
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1 answer
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Volume of region bounded

Find the volume of the region bounded by $y-x^3+1$, $y=0$, $x=0$ and $x=1$ revolved around the $x$-axis. I got $\frac{9\pi}{14}$ but the actual answer was $\frac{23\pi}{14}$. My work was to first get ...
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How many tablets can I fill into a bottle?

I have tablets/capsules at hand and I need to buy bottles to put these in. My question is, given that I know the dimensions of a particular tablet/capsule such as diameter and height, how can I ...
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Crow has to drop additional $320$ or $400$ volume pebbles to raise the water of the tank to drink?

A small treat is floating in a rectangular tank measuring 8 inches wide, 10 inches long, and 12 inches high. If the water is currently 5 inches high and a smart crow can reach at most 3 inches down ...
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How to calculate the volume of this parameterized region?

I have to calculate the Volume of a region defined by $V=\bar{x}\in \mathbb{R}^{3}:x^{2}+4y^{2}\leq z\leq 2+2x+4y$ I tried to integrate in z from $x^{2}+4y^{2}$ to $2+2x+4y$ and then using polar ...
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Area (quadratics) go negative vs. Volume (cubics) goes to infinity

Consider a typical "fenced garden" problem, which is easily represented by a quadratic, e.g., $a(w)=-10w^{2}+100w$ where $w$ is the width (or the wall, if you prefer). which peaks at $(5,250)...
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Infinite tower of cubes inside the unit sphere

Given the unit sphere it's simple to find the volume of the largest cube that can fit inside it. In the spherical cap above the cube we inscribe, again, the largest cube possible, and we keep going ...
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1 answer
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When uniformly sampling points from a high dimensional ball, intuitively, why are most of the points at the crust?

This is related to the curse of dimensionality. Many proofs can be found showing that the volume is mostly at the crust, but I don't quite understand the intuition. Consider a $d$ dimensional unit ...
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-3 votes
1 answer
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How to do volume of revolution - Integration? with cones or cylinders? [closed]

I did this question, but I'm getting the wrong answer and I don't know where I am making the mistake. I'm also uploading my working out. Could anyone help? Thanks this is the question this is mt ...
2 votes
2 answers
204 views

Deriving the formula of the volume of a sphere using integration

I'm trying to derive the formula for the volume of a sphere, using integration : $\int_{0}^{\pi r}\pi r^{2}dc$ $\pi r^{2}$ is constant, so : $\pi r^{2}\int_{0}^{\pi r}dc$ Integrating, I get only c, ...
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Deriving volume of an n-ball given the expression for n-1 ball

Given the expression for unit volume in n-1 dimention using Euler Gamma Function: $S_{n−1}=\frac{2π^{n/2}}{Γ(\frac n2)}$, I'd want to derive the expression for unit volume i n-dimentional space (which ...
2 votes
1 answer
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Using calculus to compute volume [duplicate]

There are two solid figures (I don't know what they're called just see the picture) and their radius on top ($r_1$) and bottom ($r_2$) same but height different ($h_1,h_2$). When we use calculus to ...
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Volume of $M=\{(x,y,z)\in\mathbb R^3:x^2+y^2+z^2=1,a\leq x\leq b, z\geq 0\}$

For $a,b\in\mathbb R$ with $-1<a<b<1$. Determine $\int_M dS_M$ for $M=\{(x,y,z)\in\mathbb R^3:x^2+y^2+z^2=1,a\leq x\leq b, z\geq 0\}$. How can I solve this? Is it possible to use polar ...
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What is the volume of the largest right cylinder that can fit inside a closed rectangular box of dimension $12 \cdot 10 \cdot 8$ cubic inches?

What is the volume of the largest right cylinder that can fit inside a closed rectangular box measuring $12$ inches by $10$ inches by $8$ inches? I thought we assume the radius of the cylinder equals ...
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2 votes
0 answers
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Showing in general way that three square pyramids can combine to form a cuboid having same dimensions as square pyramid Visually

I know that volume of a square base pyramid of bottom square area $S$ and height $H$ is given by $\frac{1}{3}SH $ , and volume of a square base cuboid outside having dimensions exactly that to be $SH$ ...
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Calculate the volume of a tetrahedron with vertices $(1,1,1)$, $(2,1,1)$, $(1,2,1)$, $(1,1,2)$ [closed]

Calculate the volume of a tetrahedron whose vertices are $(1,1,1),(2,1,1),(1,2,1),(1,1,2)$. I don't know how to start, do I have to find the planes?
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Understanding the reasoning behind making the approximations of rectangles and cuboids while calculating area and volumes

In this link https://www.3blue1brown.com/lessons/essence-of-calculus Grant Sanderson said if we unwrapped the ring we can get the shape as depicted below. : , ...
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1 answer
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Volume integral over dyadic: Is it true that $\int_{\mathbb{R}^3} d^3 \vec{x}\; x_i x_j f(|\vec{x}|) \ \propto \ \delta_{ij}$?

Let $f(a)$ be some function of $a>0$. Is it true that $$ \int_{\mathbb{R}^3} d^3 \vec{x}\; x_i x_j f(|\vec{x}|) \ \propto \ \delta_{ij} $$ where $\delta_{ij}$ is the Kronecker delta, and where $x_i$...
1 vote
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Immersed volume calculation

Im trying to solve the following problem : Given the ellipsoid represented by the matrix $\widehat{A}$ and knowing the coordinates of each point A,B,C,D,E,F,G,H, calculate the volume of intersection (...
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Calculating moment of inertia for a cuboid

I am trying to calculate the moment of inertia tensor for a cuboid with half edges of width $a$, height $b$, depth, $c$ and mass, $m$, which according to wikipedia, is: \begin{equation} I_c =\begin{...
2 votes
1 answer
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Calculating the center of mass of a lemniscate rotated around the x-axis?

This is a problem I have been stuck on a while, it goes as follows: A lemniscate has the equation $(x^2+y^2)^2 = 4(x^2-y^2)$. Let the part of the curve that lies in the first quadrant rotate around ...
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Setting up this volume integral

I just have a quick question on setting up a volume integral of a region bounded by $y = \sin(x)$ and $y = -\sin(x)$ for $0\leq x\leq\pi$ about the x-axis. I keep getting zero when I set it up as $x = ...
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Find the entire volume of the solid $(\frac{x}{a})^{1/2}+(\frac{y}{b})^{1/2}+(\frac{z}{c})^{1/2}=1$.

Question : Find the entire volume of the solid $(\frac{x}{a})^{1/2}+(\frac{y}{b})^{1/2}+(\frac{z}{c})^{1/2}=1$. My Attempt: After reducing it to some extent we get $V=2c\int\int_{D*} (1-(\frac{x}{a})^...
1 vote
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How to find the volume by cross section between two functions?

Consider the function $f(x) = \sqrt{x}\,$ where $\,0<x<1$ 1- This function is rotated about the $x-axis$ by an angle $\theta = \frac{\pi}{2}$ 2- Normally the volume of this rotation can be ...
3 votes
2 answers
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Volume of a solid with a circular base

Find the volume of the solid with a circular base of radius 9 and the cross sections perpendicular to the y-axis are squares. I've never solved a problem like this before. How can I go about setting ...
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Upper bound on number of lattice points of given bounded convex set in $\mathbb{R}^{n}$?

Consider the bounded (closed) convex region given by: $a^Tx = b$ , $a^Tx = b + ||a||_2$ and $ m_1 \le x_i \le M_2 $ forall $1 \le i < n$ and $m_2 \le x_n \le M_2$ i.e two parallel hyperplanes at ...
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1 vote
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How to tell if a "sliced" sphere is greater or less than a hemisphere?

I have a dataset of volumes that are roughly spheres within a volume. Kind of like a block of swiss cheese or a porous matrix like bone. In a given thickness of this volume, not all spheres will be in ...
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1 answer
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How do you interpret the polynomial equation for the volume of a 3-D Washer?

Can anyone explain what's going on in this formula for me? Formulas: [ (x^2 + y^2 ) > r1^2 ] , [ (x^2 + y^2 ) < r0^2 ] , [ z^2 < (h0/2)^2 ] Formulas with constants filled in: [ (x^2 + y^2 ) &...
1 vote
1 answer
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Volume and height of a pyramid

Given the above question. What is the height of the truncated piece. I tried creating similar triangles, one with a base equal to the height of the 5cm equilateral triangle and the other having a base ...
2 votes
2 answers
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How can Gabriel's Horn have a finite volume?

I’m a highschool student who just finished Calc AB and I’m fascinated by the concept of Gabriel’s Horn but I’m confused by the claim that its volume is finite. Correct me if I’m wrong but isn’t it ...
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Volume of $y=x^2+1; y=-x^2+2x+5; x=0; x=3$ about $x$ axis (Shell Method).

I was working on this exercise for an assignment. However, I get stuck in the following part. $ y=-x^2+2x+5 $ Complete the square $y=-(x^2-2x)+5$ $(b/2)^2=(-2/2)^2=1$ $y=-(x^2-2x+1-1)+5$ $y=-(x^...
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How to measure the "volume" of each "Matryoshka doll" in a topological space?

Let $X$ be a (perfectly normal) topological space. There is a family $\{C_i:i∈I\}$ indexed with an uncountable linear order $(I,\prec)$ such that $Ci\subset Cj$ whenever $i\prec j$. The boundaries of $...
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How to know the volume and density of the material, as well as the angle $\alpha= 42^\circ,$ to find the radius and height of the semi-cone [closed]

Is it possible, knowing the volume and density of the material, as well as the angle $\alpha= 42^\circ,$ to find the radius and height of the semi-cone [pic of semi-cone] ?
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How to find volume generated by rotating the region using Shell method

Refer to the figure below to determine the volume generated by rotating the given region around of the specified axis: a) $R_2$ around AB Using the formula $A(x)=2\pi(radius)(height)$ Why my radius ...
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1 answer
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Converting depth of irregular shape to equal given volume

I have a 3D shape with the following aerial view: Where $a\ b\ c\ d\ e$ are known integers. The shape has a depth $f$ and volume $v$. The question is, given a required volume $V$ where $V<v$, to ...
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Finding a volume using a double integral

Problem: Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola $y = 4 - x^2$ and the line $y = 3x$, while the top of the solid is bounded by the plane $...
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How to tell which side of the curve represents the equation $y=\sec^{-1} x$ and which represents $y=-\sec^{-1} x$?

If we have the $y=\sec^{-1}x$ and we want to rewrite it as $x=\sec y$ for the purpose of finding the volume using the cylindrical shell technique due to the revolution of the region bounded by $y=\sec^...
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Does this explanation for the volume of rotation about a horizontal line make sense?

I'm trying to explain how you would find the volume of rotation of $y=x^2$ around $y=19$. I am aware you can just translate the graph downwards by $19$ but the point is to try explain volumes of ...
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Volume of a rescaled Stiefel manifold

The volume of the complex Stiefel manifold (where $n>p$) $$\mathcal{S}(n,p)=\{\boldsymbol{Q}\in\mathbb{C}^{n\times p}|\boldsymbol{Q}^{\rm H}\boldsymbol{Q}=\boldsymbol{I}\} $$ is given by $$ {\rm ...
1 vote
2 answers
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Why the degree of $\pi$ does not increase

What is the underlying reason, that $\pi$ is of degree $1$ in the volume formula for an $n$-ball of radius $r$, i.e. the perimeter of circle is $$2\pi \cdot r$$ its area is $$\pi \cdot r^2$$ the ...
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2 votes
1 answer
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Calculating the volume of an arbitrary mesh

Given a mesh (vertices, edges, faces), I need to calculate the volume of the formed mesh. By volume, I mean the amount of space that a substance or object occupies. The resulting volume does not ...
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1 answer
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Areas of polygons in the the language of Riemannian geometry

I have a pedantic and somewhat vague question that has been bothering me a bit and that I was hoping for some clarification on. I'm sure it is a standard issue that there is a standard way of dealing ...
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Two volumes forms on a compact manifold

Let $M$ be a compact orientable manifold, $\alpha$ and $\beta$ are two volume forms (defined as nowhere vanishing (dim$M$)-forms) on $M$. Is it true that there exists a smooth function $f:M\rightarrow\...
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Finding Surface Area and Volume of a Rectangular Prism

So this is my work so far: Rectangle 1: Length: 10 cm; Height: 7 cm Rectangle 2 and 3: Length: 6 cm; Height: 8 cm I am getting confused on the question is that the rectangles have different side ...
1 vote
1 answer
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How to maximize volume of tube created with a piece of paper of area $1200\ \rm{cm}^2$

I am doing a math project where I have to maximize the volume of a tube created by rolling this piece of paper. The paper has a fixed area of $1200\ \rm{cm}^2$. I have been trying to use optimization ...

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