For questions related to volume.

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1answer
9 views

Volume Exponential Function

I should find the Volume received by rotating the region bounded by: $y = e^x $, $ y = 0 $,$ x = 0 $, $ x = 1 $ rotated around the x axis. I know how to find it by using the disc method but I could ...
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3answers
52 views

Volume of a 'cylinder with rounded sides'

I need to find the volume of a torus-shaped object, but it which doesn't have space between the ring. We can find the volume of the ring, but what about the inner part? PS: What is that shape ...
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0answers
14 views

Product of Riemannian manifolds and volume element

Let $X$ and $Y$ be Riemannian manifolds and consider a function \begin{align} f\colon X\times Y &\to \mathbb{R},\\ (x,y) &\mapsto f(x,y) \end{align} Now I have to integrate $f$ with ...
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1answer
40 views

Parametrization of two curves. [on hold]

I have an assigment to parametrize the edge of the volume which is given by the intersection of the two curves $x^2+y^2+z^2=2$ and $z=x^2+y^2$. I really have no idea how i can parametrize this? I know ...
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1answer
26 views

Creating an integral for finding the volume of this revolution

I need to find the volume of a solid that is created by rotating the area within the following boundaries: $y=x^3$ $y=8$ $x=0$ which is rotated over $x = 3$. I thought I had the correct integral ...
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2answers
31 views

Find volume of these solids using integration

a) The $(x>0, y< -1)$ region of the curve $y= -\frac{1}{x}$ rotated about the $y$-axis. The instructions say that one should use the formula: $V = \int 2πxf(x) dx$ I used another method and ...
2
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0answers
25 views

Volumes by Cylindrical Shells - What am I doing wrong?

I am trying to solve this exercise from a textbook: $y = x^4, y = 0, x = 1;$ rotated about $x=2$ This is my attempt at solving the problem: Shell radius: $2 - x$ Shell height: $x^4$ $a = 1$ $b = 2$ ...
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1answer
22 views

Finding volume of a solid of revolution

I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis. If I'm correct, this volume can be calculated by: Evaluating ...
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2answers
18 views

Rotational Volume

I have to find the volume of the region bounded by $ y= \sqrt{x-1} $, y=3, the y-axis and the x-axis rotated around y=5 I set up $\int_1^{10} $ $\pi((5-(\sqrt{x-1}))^2 - (5-3)^2)$dx + $\int_{0}^1$ ...
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0answers
15 views

Volume Problem in Munkres' analysis on manifolds [on hold]

I am having trouble with problem (a) of this question. I figured that the volume of $\triangle_1(R)$ is $|(\alpha(a+h, b)-\alpha(a, b))\times (\alpha(a+h, b+k)-\alpha(a, b))|$ but don't know how I ...
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0answers
23 views

Solid of revolution problem [on hold]

how do I find the Volume of the solid of revolution of $y = x^2$ rotated around the $x$-axis on the interval from $0$ to $1$ using double integrals and triple integrals
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0answers
71 views

Concavity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Is the function $f(r) = \mu(A_r)^{1/n}$ ...
0
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1answer
27 views

Why Can't I find the Volume of a Rotated Graph by Average Value Theorum?

I am wondering why I get an incorrect answer when trying to find the volume of a rotated function about the x-axis when using the Average Value Theorem. I want to find the volume of $y=\sqrt{x-2}$ as ...
1
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2answers
47 views

Reasoning Aptitude CSIR 15

A single-celled spherical organism contains $70$% water by volume. If it loses $10$% of its water content, how much would its surface area change by approximately? $3\text{%}$ $5\text{%}$ ...
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0answers
13 views

Calculate volume using flow rate and time difference in a container

So bare with me, I am a computer scientist undergrad taking part in an engineering week and this is a calculation we require for our report. So I'd like you to imagine I am filling a cylindrical ...
0
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0answers
27 views

On finding the volume using triple integral

Find the volume of the cylinder with base as disk of unit radius in the $xy$ plane centered at $(1,1,0)$ and top being the surface $z=[(x-1)^2+(y-1)^2]^{3/2}$ According to me it should be ...
0
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1answer
15 views

Finding volume using disks/washers

Find the volume of a solid obtained by rotating the region bounded by $y = \sqrt{x}$ and $y=x$ about $y=2$. I need help setting up this problem. Thank you!
2
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1answer
47 views

Finding Volume of a Cylinder with a hole (Real Life Problem)

So for my engineering coursework I need to have the masses all the parts I make. I know the density of the material I am using but I don't know the volume of this particular solid I'm about to ...
1
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1answer
24 views

Volume of revolution of the area between $\sec(x-1)$ and $\ln(x)$ for $1 \leq x \leq 2$

I am trying to compute the volume of the solid of revolution given by the rotation of the area between $\sec(x-1)$ and $\ln(x)$ for $1 \leq x \leq 2$, rotated around the $x$-axis. I have tried using ...
0
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1answer
26 views

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$$x y = 1$$ $$y = 0$$ $$x = 1, x = 2$$ About $x = -1$ This question has been bugging me for a while and I can't seem to find a way to do it. What I had done so far: $$A_1(x) = ...
2
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0answers
21 views

Volume generation by revolution [closed]

Find the volume of the solid obtained by rotating the region in the first quadrant enclosed by the curves $y = 6 − x$, $y = x^2$, $x = 0$ about $y = −3$. Can someone help me to set up integral for ...
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0answers
5 views

isotropic volume from 3D dicom data

Iam working with CTA volume having dimensions 512*512*304. How I can convert the volume into isotropic form having axial size of 256*256.I have gone through ...
3
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0answers
26 views

What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
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3answers
36 views

Finding the volume of a $\mathbb R^3$ triangle

I have a triangle $ABC$ defined with the points $A=(2.4,-5.4,6)$, $B=(0,1.1,3.2)$, $C=(-7.6,3,0)$ And I'm asked to find the volume of the solid of $\mathbb R^3$ given by the points between the plane ...
3
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2answers
47 views

Calculate the volume bounded by the surface $(x^2+y^2+z^2)^2 = x$

I need to solve: Calculate the volume bounded by the surface $$ (x^2+y^2+z^2)^2 = x $$ and not sure on how to do it. If I move to spherical coordinates, I get that the equation gives: $$ ...
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0answers
14 views

turning cartesian triple integral to spherical

I have the unit hemisphere centred at $z=1$ and a cone right under it with a point at $(0,0,0)$ and I am trying to find the volume with a triple integral. I already know that the volume is ...
0
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1answer
14 views

Revolving region bounded by $x = 16y^2 − 4y^3,\ x = 0$ about the $x$-axis, volume using cylindrical shells

Use the method of cylindrical shells to find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the $x$-axis. $x = 16y^2 − 4y^3,\ x = 0$ so first i need ...
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0answers
39 views

How to derive the volume of a sphere?

I was trying to derive the formula for finding the volume of a sphere. I tried to do it in a same manner as inspired by this answer-Where am I wrong at deriving the formula of volume of cone? ...
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0answers
21 views

Limits of the power of the double integral for finding volumes

I know that the double integral can be used to find volumes. The triple integral can also find volumes. Is the double integral limited to being only applicable for finding special cases of volumes or ...
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0answers
14 views

How to calculate the height of the cuboid tank?

Cuboid shape tank has been filled with 84 liters of water, which makes 70% of the whole tank capacity. What's the height of the tank if its length is 6 decimeters and width 4 decimeters. Can somebody ...
0
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1answer
30 views

Volume of Cone By Integration

I am trying to find the volume of a cone by integrating it in spherical coordinates, but elementary geometry suggests that my approach is incorrect. The specifications of the cone are $0\le R \le 5$, ...
4
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3answers
101 views

Where am I wrong at deriving the formula of volume of cone?

I was deriving the formula of cone volume yesterday but I was stuck at a place.My reason for asking help in Math SE is because I was not doing it by looking at any book or the internet.So,I want your ...
0
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1answer
12 views

How to use metric conversion with volume?

So this is the metric conversion - K h d b d c m I can't understand how you can convert metric squared units to volume squared units or vice versa. For example converting 3400 mm squared to ml ...
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1answer
23 views

Find the volume of the solid bounded below by $z=x^2+4y^2$ and above by $z=4x-8y-4$

I'm looking for feedback on if I did this question properly. The surfaces intersect at $x^2+4y^2=4x-8y-4$ which we can rearrange as $(x^2-4x)+(4y^2+8y)=-4$. We simply complete the square: ...
2
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1answer
35 views

Show, for a sphere, $\langle r^2\rangle = \frac{3}{5}r_o ^2$

The following equation is used as an inference but not explained in my solid state physics book (Economou, "The Physics of Solids"). I figure it's a math/geometry problem so I ask here. Can anyone ...
0
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1answer
23 views

How to use the metric conversion

I know how to use the metric conversion (K h d b d m c) when converting length to length for example 1 m squared to cm, etc. But I want to know how to use the metric conversion when converting from ...
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0answers
13 views

Best way to get combination of elements to fill given area

I am application developer and I came across interesting mathematical problem. Let's assume we are given: dimensions of space we would like to fill: a = 3m; b = 4m; set of elements: Items = { {a0, ...
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3answers
784 views

Volume of 1/2 using hull of finite point set with diameter 1

It's easy to bound a volume of a half. For example, the points $(0,0,0),(0,0,1),(0,1,0),(3,0,0)$ can do it. The problem is harder if no two points can be further than 1 apart. Bound a volume of 1/2 ...
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2answers
63 views

How to find volume of a truncated cone by slant height

If i know base radius and slant height then how i can find the volume of a truncated cone.I do some research but not able to find out how exactly these two can be used to find ...
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0answers
17 views

Finding the radius of the resulting 3D solid

If I rotate the these functions about the y-axis, how can I find the radius of the resulting solid? $$x = 2\sqrt{2y}$$ $$x = 0$$ $$y = 5$$ I am trying to find the radius of the $3$ dimensional ...
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1answer
34 views

Why average area of the horizontal slices of the conical frustum doesn't work for it's volume?

I would like to react to one of the answers on this thread (I don't have enough rep to make a comment): Use cylinder's formula for frustum (conical frustum) Where is answered: Essentially, ...
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1answer
42 views

Volume bounded between an Ellipsoid and a Cone?

I'm a bit confused about how I would be able to find the volume bounded by a cone of known theta and an oblate spheroid of b = c. I'm trying to use triple integrals for the solution, and I think I ...
0
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1answer
32 views

Calculating the volume of a rectangular hole with sloping edges

I need some help calculating the volume of a hole. The bottom of the hole is a rectangle which will have the measurements 15 by 35 meters. The depth is 8 meters. Now here is the tricky part. All four ...
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0answers
16 views

Volume bounded by sphere and two cones

Volume bounded by the 3 solids: $x^2+y^2+(z-1)^2=1\\z=\sqrt{x^2+y^2}\\z=\sqrt{3(x^2+y^2)}$ Is correct to express the integral as... ? ...
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1answer
14 views

Volume between hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and line $x = 2a$ around $y$ axis

I'm trying to calculate the volume between the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and the line $x = 2a$ around the $y$ axis using two methods but I'm getting different answers: Using ...
0
votes
1answer
11 views

Find the Volume when region is rotated around unknown vertical line

A region is bounded by the $y = 0$, $x = 1$ and $y = arctan(x)$. This region is rotated around a vertical line $x = b$, where $b > 1$. The solid formed has $Volume = 5$. To solve the problem, I ...
0
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1answer
16 views

Find the volume when a curve $16x^2-(y-8)^2=32$ is rotated completely about $y$ axis between $y=0$ and $y=16$

Find the volume when a curve $16x^2-(y-8)^2=32$ is rotated completely about $y$ axis between $y=0$ and $y=16$ Can anyone help me with this question and a little sketching of the graph might helped.
0
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1answer
27 views

How to minimize the surface area taken by a cylinder?

In my math class, we are working on Geometric Optimization problems. We have to create an equation, and then solve for one variable, in terms of another variable. Then, using an expression, we find ...
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0answers
34 views

Manifolds, where its enough to have one chart for integration

Assume a compact connected manifold $M$ is given as a subset of some $\mathbb{R}^m$. Assume we have a chart $\gamma:U \rightarrow M$ such that $M-f(U)$ (the set $M$ without $f(U)$) has zero measure in ...
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0answers
54 views

Volume generated by revolving $f(x)=\sqrt{x}$ about $g(x)=x$

Recently, I have been trying to refresh my memory about the methods of finding the volumes of solids using integration. I know it's possible to find the volume of a solid generated by revolving ...