For questions related to volume.

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0answers
19 views

Finding Volume of a solid with trigonometry [on hold]

I need help to solve this problem please. Thanks. The region between the curve $y=\sec^{-1}x$ and the $x$ axis from $x=1$ to $x=2$ is revolved about the $y$-axis to generate a solid. Find the volume ...
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0answers
6 views

Question about Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies - Kannan Lovasz Simmonovits 97

I've been trying to understand this paper: "Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies", Ravi Kannan, Laszlo Lovasz, Miklos Simonovits. On page 20 we define a measure on ...
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1answer
30 views

Find the volume of the solid obtained by rotating about the y - axis

Find the volume of the solid obtained by rotating the region bounded by $$y=2x+6,\hspace{10mm} y=0$$ about the $y$-axis. I have tried $$\int\limits_0^6\pi\left(\frac{y-6}{2}\right)^2\,dy$$ but ...
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0answers
20 views

Volume of a regiong

I am not sure if I am solving this question correctly. Consider the region bounded by graphs of y=lnx, y=0, and x=e. Find the area of the region, and find the volume of the solid formed by revolving ...
-1
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2answers
24 views

Find volume of solid [on hold]

Find volume of solid generated by revolving region in the first quadrant bounded by the coordinates axes, the curve y=e^-x and the line x=1, about y axis. Thanks
1
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0answers
28 views

Shell method resulted in a negative volume

The question is: Let R be the region bounded between the x-axis and the curve $y=x^2-4x$. Find the volume generated by rotating R about the y-axis. I used the shell method as the following ...
4
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0answers
37 views

Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
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2answers
22 views

What's the maximum volume for this cone?

What's the maximum volume a right circular cone with a slant height of 45 length units possible can have? The volume $V$ for a right circular cone is given by $V = \frac{\pi r^2 h}{3}$.
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1answer
21 views

Volume of Unusual Shape

Let $z = 1/r^n$, where $r = \sqrt{x^2+y^2}$ and $0 < n < 2.$ Note that this function has a discontinuity at the origin. Find the volume, $V(a)$, under this surface (and above the xy-plane) ...
2
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1answer
70 views

A question about multiple integral

How to compute the multiple integral $$\int \int...\int_{(D)} dx_1dx_2...dx_n, \ \ D:-1\le x_1,x_2,\ldots ,x_n\le1, -1\le x_1+x_2+\cdots +x_n\le1.$$ Thanks in advanced for your help!
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1answer
32 views

basic question - Volume of revolution of weird shape

I'm having trouble visualizing the volume that we need to calculate. The question is: Find the volume of revolution of the shape created by $y=x^2$ and $y^2=x$ revolving around the $x$ axis. The ...
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2answers
43 views

What is the area of the region bounded by $y=\ln{x}$, $y=0$, and $x=e$, and what is the volume formed by revolving this region about $x$-axis?

I could not figure out how to establish an integral formula from these questions. Consider the region bounded by graphs of $y=\ln{x}$, $y=0$, and $x=e$. Find the area of the region, and find the ...
0
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2answers
38 views

Volume of a solid with triangle base

The question is "Find the volume of the solid whose base is the triangle enclosed by x+y=3 and the x&y axes; and cross-sections perpendicular to the y-axis are semicircles" I have done many ...
1
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1answer
17 views

The volume of the solid, generated by revolving about $y = 2$ the region bounded by $y^2\leq 2x, x \leq 8$ and $y \geq 2$, is

The volume of the solid, generated by revolving about $y = 2$ the region bounded by $y^2\leq 2x, x \leq 8$ and $y \geq 2$, is (A) $2\sqrt2\pi$ (B) $28\pi /3$ (C) $84\pi$ (D) none of these. My Steps: ...
4
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0answers
52 views

Interpreting $n!$ as the volume of a $1 \times 2 \cdots \times n$ box

Q. Are there relationships or proofs that are illuminated by viewing $n!$ as the volume of a $1 \times 2 \cdots \times n$ box in $n$-dimensions? I cannot think of any, but perhaps they ...
-3
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0answers
16 views

Finding the volume of water in a cylinder lying on the curved surface [closed]

The cylinder lies not on its base but horizontal. The usual radius and length or height is given....... The water level also given..... Find the volume of water and the area of wet surface in it? ...
2
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0answers
18 views

Volume of paraboloid

Consider a paraboloid $x^2+y^2+2=z$. The task is to find the volum of the body obtained by confining the paraboloid by several planes $x=0, z=0, y=0$ and $x=3, y=3$. The zero planes cuts out a ...
0
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0answers
10 views

Region bounded by two paraboloids?

First time doing this, not sure how correct my reasoning is. $$z=12-x^2-y^2$$ $$z=4x^2+4y^2-20$$ So I guess you find where they intersect: $$12-x^2-y^2=4x^2+4y^2-20$$ $$\frac{22}{5}=x^2+y^2$$ So ...
0
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1answer
23 views

Why can't I integrate with respects to x for this volume of integration problem using shell's method?

Here is the graph of the problem, http://gyazo.com/5755dcfeea517756762943f704280d91 Here is the problem itself, http://gyazo.com/14bf36a0fa1968bcf41843e5f3510c82 This works if I integrate with ...
-2
votes
2answers
51 views

Volume and surface area of a drilled out cube (BM01 2010/11 Contest Question 2)

Let $s$ be an integer greater than $6$. A solid cube of side $s$ has a square hole of side $x < 6$ drilled directly through from one face to the opposite face (so the drill removes a cuboid). The ...
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0answers
23 views

Finding volume of solid using integration.

Find the volume of the solid obtained by rotating the region bounded by the curves: $y=x^8$ and $y=1$ about the line $y=6$. I tried to use the washer method but then I cannot find two volumes that I ...
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1answer
259 views

How do you find the cross sectional area of a Tetrahedron?

How is vector's related to this question? If so, how can you use the vectors? I understand that its a triangular pyramid. But how can you show the cross sectional area for any generalised height? ...
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0answers
17 views

Find largest regions bounded by a set of planes

Suppose we are given a set of planes that partition the unit cube into a large number of regions. Is there a computationally efficient way to find the region with the largest volume?
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1answer
32 views

Find the volume V of the solid bounded by the cylinder $x^2 +y^2 = 1$, the xy-plane and the plane $x + z = 1 $.

Find the volume V of the solid bounded by the cylinder $x^2 +y^2 = 1$, the xy-plane and the plane $x + z = 1 $. Hi all, i cant seem to get the correct answer for this question. The answer is $\pi$ ...
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0answers
9 views

Using Washer Method to find volume of solid

How do you find the outer radius of the the solid bounded by y=4-x^2 and y=1 that rotates around x=2?
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1answer
24 views

Volume of solid with known cross-section

To find volume of solid with $x=y^2$ and $x=9$ perpendicular to $x$-axis and cross-section to be taken is triangle with $h=\frac b4$. I am confused in imagining the triangle. $$\text{Volume to me} = ...
0
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1answer
13 views

Volume /mass / density problem

If I have a solution that has a volume of 32.5mL, a mass of 40.0g, and a density of 1.23 --- how do I solve for 10mL, 15, 20mL, etc of the same solution? Thanks!
1
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0answers
69 views

Four holes in a sphere

Supposing that the sphere $x^2+y^2+z^2=9 $ has four central holes of unit radius drilled through it. The radial holes are directed towards the sphere center starting from hypothetical regular ...
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0answers
24 views

removal rate of two cylinders

Suppose the following situation where one of the cylinders moves towards the other one, along the x-axis. The axis of the moving cylinder is parallel to the y-axis and let $\alpha_1$ be the ...
0
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2answers
22 views

Calculus- Shell Method Problem - find the volume of the solid when the region $y^2=x, x=0,$ and $y=1$ is rotated around $x=-3.$

Since the problem uses $y^2=x$, I first assumed that the element must be horizontal (parallel to the $x$-axis). However, the bounded region has all $y$ values greater than $0$, so I could also use a ...
2
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1answer
27 views

Volume of a compact set, not necessarily convex

Looking through my lecture notes, I came across the notion that if a set $X\subset \mathbb{R}^n$ is compact and convex and $vol(X)=2^n$, then by choosing an $0<\epsilon <1$, then $X\subsetneq ...
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0answers
13 views

Smallest diameter for convex polytopes with a given volume

Let $K\subset\mathbb R^n$ be a full-dimensional, bounded, convex polytope formed by the intersection of $m$ half-spaces. Is there a nice lower bound for the diameter $D(K) = \sup_{x,y\in K}\|x-y\|_2$ ...
1
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1answer
52 views

Volume integral over a wedge shaped cylindrical region

i'm having trouble picturing this and figuring out what the limits of my integrals should be. Calculate the volume integral of $f(\rho,\phi,z) = 2z$ over the wedge shaped cylindrical region, where ...
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0answers
47 views

How to calculate the critical density estimation for “continuum” percolation model in “3D space” when we have “spatial correlation”?

I want to approximately estimate the critical density (lower bound for density) of balls in a cube to make sure that the upper and lower surfaces of the cube will be connected to each other through ...
0
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1answer
9 views

Finding Desired Rate of Change that Results in Same Volume

Let's say we have an elliposoid given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$. The volume is given by $V = \frac{4}{3} \pi abc$. If at a certain moment in time $a = ...
0
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2answers
39 views

Volume of a Solid, $x^2 - y^2 = a^2$

The question is Find the volume of a solid rotated around the y axis, bounded by the given curves: $$x^2 - y^2 = a^2$$ $$x = a + h$$ I am lost by the number of variables in this question ...
0
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2answers
37 views

Volume - Integration Question Help

So basically, with this question I have answered parts a) and b), but I am stuck in relation to part c). I am not exactly sure what the borders of my integration should be, but I figured with the ...
4
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1answer
50 views

Why, intuitively, do different shapes with the same surface area have different volumes?

This is something that's always bothered me. I am well aware that you can easily see why this is the case with math. I mean, even in the 2-D case, take a square with side length $1$, and it has a ...
0
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1answer
47 views

Volume of Intersection of cylinders (different radii)

I want to derive a formula for the area of the intersection of two rigth cylinders with different radii. To get an idea I attached a sketch. My idea is to determine the borders of $x$ and $z$ in ...
2
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4answers
77 views

How does Volume work with integration?

Using a cross section suppose, as described here: Area formula Paul Notes Suppose this is: $y = f(x)$. He says the volume is: $$\int_{a}^{b} A(x) dx$$ But how does area over that interval ...
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1answer
36 views

Show that it is the volume

A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$. Let $\overrightarrow{n}$ an unit vector perpendicular to the plane. Show that $\overrightarrow{v} \cdot ...
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2answers
31 views

Find the volume of this improper integral?

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by $f(x)=\sqrt{\frac{(x+1)}{x^3}}$ and the $x$-axis on the interval $[1,\infty)$ is resolved ...
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3answers
40 views

Calculus Cross-Sectional Volume

I missed two classes in calculus and we're on a subject that I do not understand at all. If someone could just walk me through this problem I could probably begin to comprehend the rest. The base of ...
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1answer
55 views

Volume and center of mass of a drilled out hemisphere

Through a homogeneous hemisphere with radius $R$ a hole is drilled with radius $R_0=R/2$ centrally so that the hole axis coincides with the hemisphere's axis of symmetry. (a) How much is the ...
0
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1answer
61 views

Volume of revolution - shell method

A mould for a circular fish pond is made by rotating the region bounded by the curve$ y = 2-\cos^2 x$ and the $x$-axis between $x = \displaystyle -\frac{\pi}{4}$ and $x = \displaystyle \frac{\pi}{4}$ ...
0
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1answer
38 views

Maximization: Volume of paraboloid within cone?

Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the ...
0
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1answer
29 views

time until gas is depleted cylinder

If one has a $10$ liter (water volume) tank of gas at $15$ Mpa, then how long until gas is depleted if gas is exiting the cylinder at a rate of $15$ liters per minute? ATM i have used ideal gas law, ...
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0answers
29 views

Volume of an union of ellipsoids

I am looking for the volume of the following ensemble, which is the union of $K$ ellipsoids: $$ \left\{ x\in\mathbb R^d : \sum_{k=1}^K \frac{\exp\left[- (x-m_k)^T \Sigma_k^{-1} (x-m_k)/2 ...
0
votes
2answers
40 views

Area of parallelogram and volume of parallelpiped using cross-product

I just started my vector calculus course a couple of weeks ago and I understand that the area of a parallelogram is Area = |A x B|, where A and B are position ...
0
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1answer
37 views

Liquid height in an inverted half-filled cone.

I have a question that I have been puzzling over for a couple of hours now, but I can't seem to understand. "A right circular cone is filled with liquid to a depth of half its vertical height. The ...