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Questions tagged [volume]

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

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1answer
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find equal 2nd and 3rd dimensions of rectangular prism when given volume and one side

I am interested to know how to calculate the length and width (which should be equal) of a rectangular prism knowing only the overall volume and the height. So for example ...
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1answer
22 views

Volume of rotating area around the $x$-axis.

I have an area $R$, which lies beneath the curve $y = 9 - x^2$ and over the line $y = 5$. What is the volume when I rotate $R$ around the $x$-axis? This is what I've done so far: First I wanted to ...
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3answers
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A solid sphere $x^2+y^2+z^2 \le 1$ is cut into two parts by a plane $z=\frac{1}{2}$.

A solid sphere $x^2+y^2+z^2 \le 1$ is cut into two parts by a plane $z=\frac{1}{2}$. Find the volume of the smaller part. I have no idea how to approach this. It will be helpful if you give me the ...
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1answer
12 views

Finding the limits of integration for a triple integral.

I am trying to compute a triple integral of the form $\int_Vf(x,y,z)dV$ where $f(x,y,z)=8xyz$ for where our limits are $0<x<y<z<1$. Does this mean we have limits of integration at: $dz: \ ...
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1answer
27 views

Do all compact manifolds have finite volume?

Let $M$ be a compact $k$-dimensional manifold embedded in $\mathbb{R}^d$. Does $M$ being compact imply that the $k$-dimensional volume of $\operatorname{vol}_{k}(M)$ is finite?
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How to prove cylindrical shell method using Darboux Integral

So, I am trying to prove that the volume calculated using cylinder shell method is $$\int_b^a2\pi f(x)dx$$ So basically we know that, $$f(u_i)(\pi x_{i}^2-\pi x_{i-1}^2)\leq\Delta V\leq f(v_i)(\pi x_{...
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1answer
17 views

Volume of the solid from rotating four curves

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=5+1/(x^2), y=5, x=3, x=6; about the x-axis. I'm not sure how to solve this ...
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2answers
35 views

Volume of the solid from rotation

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=3x^2, x=1, y=0$, about the x-axis.
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0answers
27 views

How to derive the volume of the region left when a plane cuts a solid elliptical cylinder?

I need to be able to calculate the volume of an elliptical cylinder that is cut by a plane. It is similar to slicing a cylinder and finding the volume, only this time with an elliptical cylinder. I ...
2
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2answers
45 views

Why is this way of deriving surface area of sphere wrong when a similar method can be used to derive volume?

Suppose a sphere with radius R is centered at the origin, whose cross section is as follows (R is the constant radius while r is variable): then its volume can be easily calculated: $V=\int_{-R}^{R}\...
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1answer
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Expressing the volume in cylindrical coordinates which is obtained by intersecting the surfaces

If $C$ is the region obtained by intersecting, the surfaces $z=x^2+y^2$ and $z=2-x^2-y^2$. Express the volume of $C$ as the triple integral in cylindrical coordinates. My try: $x^2+y^2=2-x^2-y^2$ ...
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1answer
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Finding the volume of the tetrahedron using triple integrals

$D$ is the tetrahedron bounded by the coordinate planes and the plane $3x+3y+z=3$, then express the volume of $D$ as a triple integral. Mt Try: The z-limits are $0\le z\le3-3x-3y$ If $y$ is the ...
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0answers
39 views

Volume by disk or washer

The volume of the solid obtained by rotating the region enclosed by $y=x^2, x=y^2$ about the line $x=−5$ can be computed using the method of disks or washers via an integral I am doing it like $...
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0answers
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Volume of a cone that is hollow in the middle

I've got this problem: I have a cone (cone $A$) with a radius $8$ and a height of $12$. Now lets say that cone $A$ has a thickness of $2$. This thickness can be found from the bottom and inward and ...
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3answers
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The volume of a child's model plane is $1200~\rm{cm}^3$ & that of a full size plane is $4050~\rm{m}^3.$ Find the scale of the model in the form $1:n.$

The volume of a child's model plane is $1200~\mathrm{cm}^3.$ The volume of a full size plane is $4050~\mathrm{m}^3.$ Find the scale of the model in the form $1:n.$ I thought of first converting the $...
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0answers
18 views

Find the volume of the region bounded by paraboloid.

Find the volume of the solid in the first octant bounded by the paraboloid $z=36-4x^2-9y^2$. As- In the first octant $z$ varies from $0$ to $ 36-4x^2-9y^2$. Further $y$ varies from $0$ to $2\sqrt{1-...
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1answer
21 views

Volumetric Computation — Bounded in 3 Dimensions

I need to compute the volume of the 3-D domain bounded below and by the sides by the positive orthant $x\ge0, y\ge0, z\ge0$, bounded from the top by the paraboloid $z=x^2+y^2+9$ and bounded on the ...
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0answers
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Integrating the other axis without rearranging equation

I've learned integration at school and furthermore how to find the "solid of revolution" using the disk method. In this method, we may use the equation $V=\pi\int^{b}_{a}|\ f(x)^2-g(x)^2|dx$ to ...
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1answer
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Intuition on Double Integrals

Frequently, I am met with problem that ask to evaluate a double integral over a bounded region. For example, evaluate the double integral $$\int\int_R 2x\cos(y)+3 \space dA$$ over the region $R$ ...
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A combinatorial proof of determinant as the hyper-volume bounded by vectors?

There are many proofs that the determinant of a 2x2 matrix is $ad - bc$ which is the area of a parallelogram bounded by the row (or column) vectors of the matrix. They come in many forms: plain ...
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1answer
27 views

Find the volume

Find the volume formed by rotating the region enclosed by $x = 36y$ and $y^3 = x$ with $y \ge 0$ about the y-axis . My try : We can conclude from the conditions that $x=0 $ , $y = 0$ , $x = 6^3$ and ...
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0answers
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Volume of polyhedron similar to prism but with different bases

Imagine a polyhedron similar to a prism, with parallel but different bases. Let the bases have the same number of sides, so that every vertex on a base is connected to exactly one vertex on the other ...
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1answer
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Find the volume of the surface.

Find the volume of the solid in $xyz$-plane bounded by $y=x^2,y=2-x^2,z=0$ and $z=y+3$. I have found the answer $\frac {13 \pi} {6}$. Is it correct at all? Please verify it. Thank you very much. I ...
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2answers
47 views

How do I calculate this volume using double integral?

I tried solving it by taking the $z = 4-x^2$ as the integrand then integrating w.r.t $x$ from $-2$ to $2$. Then w.r.t $y$ also from $-2$ to $2$ since the boundary equation was $x^2 +y^2 = 4$ which ...
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0answers
34 views

Find Volume of Ellipsoid Bound by Two Non-Parallel Planes

I'm trying to find the volume of a wedge of an ellipsoid. The ellipsoid is the standard form: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \leftarrow(Eqn 1)$$ The first bounding plane is the $...
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1answer
19 views

Rotating a figure around x-axis whose area is given

Let's say I have a 2d-figure above the x-axis and I'm already given it's area (so I don't have to integrate). All of this figure touches the x-axis, so there's no hole in the middle when rotating How ...
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1answer
49 views

A silver cube has edge lenghts of 5 cm each. A stamp mark “800” indicates that…

the cube is made of 800/1000 pure silver. Assume that silver has a density of 10.5 g/cm3 and a price of 0.68Euro/g. Calculate the value of the silver partion? Haven't studied math since 15 years so I'...
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0answers
66 views

Find the volume below the surface $z=x^2+y^2$ above the plane $z=0$ and inside the cylinder $x^2+y^2=2y$.

Find the volume below the surface $z=x^2+y^2$, above the plane $z=0$ and inside the cylinder $x^2+y^2=2y$. I'm getting the answer as : $$ 2\int_{0}^{2}\int_0^{\sqrt{2y-y^2}}(x^2+y^2)dxdy $$ but I'm ...
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1answer
42 views

Help with a simplification

I'm trying to solve what I thought was a fairly simple integration problem where I need to find the volume of something, but I've gotten stuck. I'm using the "disk/washer" method, and the integral I ...
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0answers
39 views

Double Integration Applied to Volume

I am trying to solve a question from Schaum's Calculus but I have been unable to. Find the volume common to $\rho^2 + z^2 = a^2$ and $\rho = asin(\theta)$. I have tried several things and the one ...
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0answers
27 views

Volume of a solid as the limit of a sum and as an integral

I have this expression: $(x^2+y^2+z^2)^2=x^3+\frac{3}{10}x(y^2+z^2)$ I would like to calculate the volume as the limit of a sum and also as an integral. Can you also point out the difference ...
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7answers
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Paradox about the volume of a cylinder

Trying to apply Cavalieri's method of indivisibles to calculate the volume of a cylinder with radius $R$ and height $h$, I get the following paradoxical argument. A cylinder with radius $R$ and ...
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1answer
21 views

Volume using cylindrical coordinates

I have to find the volume of the solid which base is bounded by $$x^{2}+y^{2}+2y=0$$ and it's bounded, above, by the surface $$z=4-x^{2}-y^{2}$$ I tried to use cylindrical coordinates, where $$x=r\...
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1answer
13 views

A Quick Way to Find the Volume of a Dodecahedron from Surface Area

The surface area of one dodecahedron is 9. A similar dodecahedron has a surface area of 16. What is the ratio of the volume of the first dodecahedron to the second? I need a quick way to solve this, ...
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0answers
17 views

Solids of Revolution Using Given Argument

We have seen 2 techniques for finding the volume of solids of revolution: the disk/washer method and the method of cylindrical shells. It would be ridiculous if these 2 techniques gave different ...
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1answer
45 views

Linear transformation of Gram determinant

The Gram determinant of the vectors $\mathbf x_1,\mathbf x_2,\dots,\mathbf x_k \in \mathbb R^n$ is defined as: $$\Gamma (\mathbf{x}_1, \ldots ., \mathbf{x}_k) = \left|\begin{array}{ccc} \mathbf{x}...
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0answers
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Lower bounds on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
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0answers
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'Jacobian' of QR decomposition of a rectangular matrix

I want to calculate the volume of real Stiefel manifold $V_{k}(\mathbb{R}^N)$ . $$ V_{k} (\mathbb{R}^N) = \{ H \in M(N, k, \mathbb{R})| H^{T}H = I_{k} \} $$ ((^T) denotes transposed matrix. $M(N, k, \...
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1answer
25 views

Volume of $x^2+y^2\leq4,\quad z=2+x^2+y^2,\quad z\geq-1$

Find the volume of the solid defined by $$x^2+y^2\leq4,\quad z=2+x^2+y^2,\quad z\geq-1.$$ I found the intersection of surfaces: $$S\equiv\begin{cases} x^2+y^2&=4\\ 2+x^2+y^2&=z\\ z&=-1 \...
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1answer
61 views

Precalculus in a Nutshell, Geometry, Question 11

"An equilateral triangle and a square are inscribed in a circle, with a side of the triangle being parallel to a side of the square. The entire figure is revolved about that altitude of the triangle ...
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0answers
12 views

surface of revolution from known volume

I would like to know how many cement I need to construct a water tank if I know the volume of the water. For example if the consumption volume is 20.5m3 and my surface of the revolution is an ...
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2answers
36 views

Volume of Cone in Pyramid by Slicing/Integrating

This is my first time making a post on here but I am currently struggling with a problem. I am unsure of how to proceed with this problem as I have only really worked with volumes of revolutions ...
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1answer
35 views

Why is the volume of an $n$-cube definable?

According to Wikipedia, the definition of Volume is given as "Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, ...
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1answer
26 views

Volume of a section of a sphere

A section of a sphere is described by $0 ≤ 𝑟 ≤ 2$, $0 ≤ 𝜃 ≤ 90°$, $30° ≤ 𝜑 ≤ 90°$. Now, just by using simple symmetry, I can see that the volume of this section will be less than 1/8th of the ...
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1answer
118 views

Volume of a cuboid cut by a sphere tangent to 4 of its edges and the 2 faces of the cuboid

As you can see from the title my problem is rather complex, so I'll try to break it apart and explain it step by step. My original problem is to find the formula of the volume of a triangular prism (...
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1answer
42 views

Is it possible to integrate this function without using spherical coordinates?

I am looking to solve the integral ${\int\int\int{\rm e}^{-\sqrt {{x}^{2}+{y}^{2}+{z}^{2}}}} dxdydz$ My boundary conditions are the boundaries of a cube (not nicely aligned with the origin), so I'd ...
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2answers
24 views

Volume of Revolution - 3d object

Recently we were given an assignment in which we have to model the cross-section of a 3d, symmetrical object utilising functions. Then we must find the volume of revolution of the object. I need ideas ...
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1answer
30 views

Volume of a solid defined by two curves rotating around $x$-axis.

I'm trying to determine the volume of the solid that is obtained by rotating around the $x$-axis the region limited by $y=2x$ and $y=x^2$. I'm trying to follow James Stewart's "Calculus. Single ...
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1answer
48 views

Functions other than $1\over x$ that generate surfaces akin to Gabriel's trumpet

When I took college calculus (more decades ago than I care to admit), we were introduced to a function that would create a surface of revolution having a finite volume but an infinite surface area. In ...
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0answers
26 views

Volume of a $d$-dimensional Torus.

Is there a known equation for the volume of a solid $d$-dimensional torus, that is $T = S^1_{r_1} \times B^{(d-1)}_{r_2}$, where $S^1_{r_1}$ is a circle of radius $r_1$ around which we revolve a $(d-1)...