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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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Volume of a hollow cone using triple integral?

I want to directly find out the volume of a hollow cylinder instead of subtracting the smaller volume from the bigger volume...Considering both the outer cone and inner cone to have the same cone ...
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I need a triple Integration setup to this problem

Write an iterated triple integral in the order dzdydx for the volume of the region bounded below by the xy-plane and above by the paraboloid $z = x^2 + y^2$ and lying inside the cylinder $x^2 + y^2 = ...
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Finding the volume by triple integration

Find the volume of the solid bounded by the planes $x = 0, y = 0, z = x, z = 2,$ and $y = 4– x^2$ in first quadrant? what's the right triple integration set up for finding the volume of this? only ...
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Volume of a solid by triple integration

how can I solve this problem with triple integrations I have tried this but I don't get the true value $\int_1^3\int_1^2\int_0^{x^2y^2}dzdydx$ Find the volume of the solid bounded by the planes $z = ...
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How can I create a triple integration for this question

How can I solve this Problem using triple integration? Find the volume in the first octant bounded by the planes $x+z=1$, and $y+2z=2$? I don't want the whole solution I just want the triple ...
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triple integration to find volume

how can I solve this problem I already have the final answer but I don't know how to solve it , the answer for this problem is $(V=8(pi-1/3)$ I tried solving it with this: $\int_{-2}^2\int_{-\sqrt{4-...
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Area under curve $\frac{1}{x}$ is infinite, volume of revolution $\frac{1}{x}$ is $\pi$?

Stumbled across this weird phenomenon using the equation $y = \frac{1}{x} $. Surface Area: When you calculate the surface area under the curve from 1 to $\infty$ $$\int_1^\infty \frac{1}{x}dx = \...
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Diluting 91% alcohol to 70% as a cost savings exercise

I'm trying to solve this and I'm not entirely confident as it has to do with dilution. This is all about trying to find the most cost effective way to obtain 70% isopropyl alcohol for cleaning. I can ...
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Volume of the fluid in the upward direction

Let $\mathbf{V}$ the velocity vector of a fluid particle at the point $(x,y,z)$ in a steady-state fluid flow. $$\mathbf{V}=x\hat{\mathbf{i}}+y\hat{\mathbf{j}}+3z\hat{\mathbf{k}}$$ Let $S$ be the ...
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Triple Integral in spherical coordinate

$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$; Hi guys, I don't quite get which region is this, is it that ...
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Calculating a volume and the centroid of a set $S$

$S$ is the region that is enclosed by the unit sphere $x^2+y^2+z^2=1$ and the cone $z=(x^2+y^2)^{1/2}$. So I decided to use cylindrical coordinates first: Define $G(r,\theta,z)=(rcos\theta,rsin\...
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Find ratio of the volume of two cone

Given two sector ABC and PQR, $\angle A=2\theta$, $\angle P=3\theta, AC=2r, PR=3r, $ both sectors are folded into a right circular cone, find the ratio of the volume of two cone. I am having ...
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mass density volume Integration

I have a volume $V$ bounded by the following equations: $ x^2 + y^2 + z^2 = 1 $ $ z^2 = (x^2 +y^2) {\sqrt2} $ and I have to find out the mass of the volume given that: $\rho(x,y,z) = z$ where $\...
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Need help with the following question about finding volume of improper integral

Find the exact volume of the solid created by rotating the region bounded by $f(x)=\frac1{\sqrt x\ln x}$ and the $x$-axis on the interval $[2, ∞)$. State the method of integral used. My issue ...
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Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$

This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same. $\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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1answer
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How many rain drops in a storm?

According to Rain Wiki : Heavy rain — when the precipitation rate is > 7.6 mm (0.30 in) per hour,[106] or between 10 mm (0.39 in) and 50 mm (2.0 in) per hour According to USGS Water Science School ...
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Finding volume of solid under $z = \sqrt{1-x^2-y^2}$ above the region bounded by $x^2 + y^2-y=0$

Find the volume of the solid that is under the hemisphere $z=\sqrt{1-x^2-y^2}$ above the region bounded by the graph of the circle $x^2 + y^2-y=0$. I solved this problem using limits of integration $...
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Volume of hyperbola revolved about the y -axis

I'm trying to calculate the volume of the solid formed by revolving the hyperbola ${x^2} - {y^2} = 1$ bounded by $x=1$ and $x=3$ about the y axis, however I do not know if I'm going about this the ...
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Volume of cuboid with unequal heights [closed]

Can you find the volume of a cuboid which has different lengths for every edge? Is there a way to calculate it accurately or do I need to use numerical methods?
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Studying spherical coordinates

$(1)$ Please suggest some books regarding the fundamental studies on surface and volume integrals in spherical coordinates. $(2)$ Are there any books dedicated to only elementary calculus of ...
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Approximate volume of an ellipsoid using a random 'length' of the object

I am a graduate student in an evolutionary biology program. We're looking to estimate the effects of random mutations of egg size in the nematode C.elegans. The eggs can be approximated as a football ...
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1answer
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How do I find the maximum volume of an A4 piece of paper using the isoperimetric inequality?

Through my research on this site, I recently stumbled across a post where it was stated that the maximum volume for a sheet of A4 paper (210mm x 297mm) is < 2.072 l. This was found to be so using ...
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Why is the singularity ignored?

In this article "Reflections on Maxwell’s Treatise", Section 4.2, it says: He replaces $\mathbf{m}$ with a volume element of magnetization $\mathbf{M}\ dV$ , integrates over $V$ , and lets the same ...
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Calculus- Volume of a Solid Revolution about the Y-axis

It asks to find the volume of the following curves given that: y=x^3, x=0, y=8 given that I already have 1 x-point, I have to equate for the other one 8=x^3, where x=2. now by plugging in the ...
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Maximise right circular cone volume with fixed surface area using inequalites

This is a question inspired from What is the relation between height and radius of base of a right circular cone when its surface area is given and volume is to be maximum? In the linked post, OP has ...
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Find the Surface Area of Revolution

Question: Let g(x) = (1 - 2x)(x - 3), and let A be the region enclosed by y = g(x), x = 1, x = 2, and y = -1. If we revolve the region A about x-axis we obtain a solid. i. Find the volume of this ...
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Volume and surface area of region by a curve and line

Let $g(x) = (1-2x)(x-3)$, B be the region enclosed by $g(x)$, $x = 1$, $x = 2$ and $y = -1$ after revolve the region B about x-axis, we have a solid. The volume of solid and the total surface area is?...
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Volume of revolution found by rotating the region bounded by $x=y^2$ and $x=1-y^2$ about the line $x=3$

Find the solid of revolution obtained by rotating the region bounded by the curves $x=y^2$ and $x=1-y^2$ about the line $x=3$. To solve this problem I tried using the washer method with respect to $y$...
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Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$

Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$ Not sure how to proceed
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Use the Jacobian to find the volume element corresponding to dV=dxdydz in the transformed integral.

Given $x,y,z$, I calculated the $3\times3$ matrix determinant. My assignment wants it in the form $$dx\,dy\,dz = \text{__________ }du\,dv\,dw$$ Would I just put the Jocobian into this? $$\begin{...
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What is the volume of the cone resulted when the first-quadrant portion of the graph of $y = −2x + 4$ is revolved about the $y$-axis? [closed]

I am having trouble with visualizing this problem: QUESTION: What is the volume of the cone resulted when the first-quadrant portion of the graph of $y = −2x + 4$ is revolved about the $y$-axis? ...
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Medial axis of symmetric solid & volume

I have a problem and would appreciate some pointers. In short, I have a solid which is produced by revolving a 2D polyline around an axis, thus producing a solid of revolution. Now: 1) Is there some ...
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1answer
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compute the volume of the solid E

The points (x,y,z) of this solid projected on the plane z=0 are in the parallelogram (2,1),(6,-1),(7,0),(3,2) and $\le z \le e^{x+2y}$. $$\int_{2}^{7}dx \int_{-0.5x+2}^{-0.5x+3.5}dy \int_{0}^{e^{x+2y}...
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Given two functions, find the volume of the solid of revolution when the region bounded by the functions is rotated about the line y=-2.

I am expected to use Mathematica to complete this task. Given the functions $f(x)=\arctan x$ and $g(x)=2\cos x$, find the volume of the solid of revolution when the region bounded by $f$ and $g$, ...
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Compute the volume of a cube not axis-aligned [closed]

I have objects in 3D which have 8 corners and most of the time they look like cubes. I need to compute the volume of those objects. Since they are not axis-aligned I can't use the standard formula $V =...
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Intuitive Explanation to Pappus Theorem

Pappus's theorem is as follows: First theorem: The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on ...
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Greatest volume of sphere inscribed in a cone with given Lateral area

The Lateral area of a right circular cone is $S$. Fine the Greatest volume of Sphere that can be inscribed in it. My try: Given that $$\pi rl=S$$ Let the radius of the sphere inscribed in it be $R$ ...
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Volume of solid revolution about $x$-axis limits

Question: Find the volume of solid formed by revolving the region bounded by the given function $f(x)$: $$ Y= 3 – x ;$$ about the $x-$axis. How do I find the upper and lower limits of integration ?
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2answers
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An Integral Error

I was studying the derivations for the volume and surface area of a sphere . One derivation , for the volume of the sphere is the disk-method . A circle of radius $r$ is considered , centred at ...
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1answer
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How to find the ratio of height/radius such that surface area of a cone is minimized?

I know that the extrema should be taken at the derivative of $V$, where $${V=\frac{\pi}{3}\cdot r^2\cdot h}$$. But I could only find where it maximizes. That's at h/r is equal to 2.
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Volume of water in container

Volume of a container is $\frac{4 \pi}{3}$. Water can flow in and out of container.The volume of water in container is given by: $g(t), 0 \leq t \leq 4$, where $t$ is time in hours and $g(t)$ is ...
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conversion question with liters

A tank has a square base of 3 m by 3 m. Water is poured into the tank at a constant rate. The mass of water in the tank is observed to increase at a rate of dm/dt = 500 kg/s. You may assume ...
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What volume is enclosed by $k$ evenly-spaced, overlapping American footballs whose axes are diameters of a unit sphere?

Take $k \in \Bbb N$ intersecting American footballs and configure them inside a unit sphere such that each football touches two opposite ends of the sphere. Each of the shapes are spaced evenly apart. ...
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Asymptotic behaviour of the intersection volume of balls with the same radius

Let $x,y \in \mathbb{R}^n$ be two fixed points. Is there an easy proof of the fact that $$A(r):=\frac{ \text{Vol}(B(x,r) \cap B(y,r))}{\text{Vol}(B(x,r))}$$ tends to $1$ when $r \to \infty$. I ...
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How to find limits of integral to find volume?

Find the volume generated by the plane region, in the first quadrant, bounded by the graph of the function $ y=\sqrt{9-x^2} $ sbout the y-axis. I know how to solve it using the formula but how do I ...
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Looking for where I went wrong: Finding the volume of a solid that lies within both a cylinder and sphere

I'm currently working on this question: Find the volume of the solid that lies within both the cylinder $x^2+y^2=1$ and the sphere $x^2+y^2+z^2=4$. I decided to use polar coordinates so that the ...
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How to find volume of the region $\{(x,y,z)|\,0 \le (x-1)^2+y^2 \le z(1-z)\}$?

I need to find the volume of the region : $$R=\{(x,y,z)| \space 0 \le (x-1)^2+y^2 \le z(1-z)\}$$ I don't understand the region. Is it the volume outside the cylinder and inside a sphere? Can you ...
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1answer
34 views

Triple integral : volume

I need to calculate the volume between $x^2+y^2\le z^2-1$ and $2x^2+y^2+z^2\le 2$. So It's a hyperboloid of two-sheets intersected with an ellipsoid. their intersection leads to: $3x^2+2y^2=1$ which ...
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Cylinderial Shells:Volume of a bounded Region

I have to find the volume of the bounded region by the following functions: $$y=\frac{1}{x^3}$$ $$y=0$$ $$x=1$$ $$x=2$$ $$\mathbf{Revolving\ around\ the \ axis:}$$ $$x=-1$$ The method ...
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1answer
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How to calculate volume of a right circular cone's hyperbola segment given position of slice?

PROBLEM: I am working on calculating volumes of geometric solids. All shapes have been pretty basic until now. I am bewildered on how to attack the problem of calculating the volume of a slice of a ...