For questions related to volume.

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1answer
42 views

Rotated arc in $\mathbb{R}$ [on hold]

We have got the arc $$A = \{(x,y) \in \mathbb{R} \mid x^2 + y ^2 = R ^2, 0 \leq x \leq R, 0 \leq y \leq R\}$$ and $R$ is positive real number. What is the area of ​​rotational figure obtained if ...
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2answers
25 views

Calculating the volume of a cone given the surface and $s$

I've been struggling with this for so long and I never got a chance to ask my teacher how to solve it. If the surface of the cone is $360\pi$ and $s = 26 \text{cm}$, calculate the volume of that ...
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1answer
38 views

Calculus question that is driving me nuts. Rotation volume. [on hold]

Find the volume formed by rotating the region enclosed by: $y=\sqrt[5]{x}$ and $y=x$ about the line $y=25$.
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1answer
34 views

How to calculate volume from surface area

I wanted to see is there a formula or method to calculate Volume of a 3D geometry, exactly Cube, from total surface area??? Plz help me in this Thanks a lot….
3
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1answer
31 views

Volume of a solid(between two planes)?

A solid lies between planes perpendicular to the y-axis at $ y=0$ and $y=1$. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola ...
2
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2answers
39 views

Shell method to find the volume of a solid?

Region bounded by $y=3x-2$, $y=\sqrt{x}$, and $x=0$ about the $y$-axis. I have been doing the washer method for all of my problems up to this one, and cannot seem to find a good resource to help guide ...
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0answers
26 views

Triple intergrals with Volume [on hold]

This question is a continuation of number 2 from the questions requiring deeper understanding of this homework. Suppose that you do not know the values of $a, b$, and $c$. Generalize what you did in ...
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0answers
13 views

Volume of parallelepiped gets smaller when using projection vectors

Given a Euclidean Space R and a subspace R' (of dimension $\geq$m), consider vectors $x_1,...,x_m \in$**R**, and let $V[x_1,...,x_m]$ mean the volume of an m-dimensional parallelepiped formed by those ...
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2answers
27 views

How to find the halfway point of a volume of a solid [closed]

How can I calculate the x coordinate which marks exactly half of the volume of a solid generated by the following region? y=(x)^1/2, ...
-1
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1answer
26 views

How to find the volume of revolution around a vertical line x [closed]

How can I evaluate the volume of a solid generated by the following lines using the washer method: $y=x$, $y=0$, $y=4$. Rotated about $x=5$. I have tried to find the outer radius of $5-x$ and the ...
0
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1answer
34 views

How to find the volume of revolution around a line y [closed]

Using the washer method how can I calculate the Volume of a solid generated by the lines y=x,x=0 ...
0
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1answer
18 views

Volume of a parallelepiped depending on $\lambda$

I've got a relatively simple calculus problem here but it has an unknown variable that I am not sure how to deal with. Find the volume of the parallelepiped depending on $\lambda$ with; $a = ...
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0answers
13 views

Functions to fill spheres at certain points in a cylindrical volume?

I know how to find how many spheres of a given radius can fit into a cylinder. But, I also want 2 know how to fill them throughout points in the cylinder. To give an example I have spheres of a ...
0
votes
1answer
30 views

Volume of a solid in R3

How can I find the volume of this field? : $$ G=\{\left. (x,y,z) \, \right| \, x^2+y^2+z^2 \le 16 \wedge 1 \le x+y+z \le 2\}. $$ Can anybody help me? Thanks.
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2answers
13 views

Volume integral of a gradient

Assuming we have a scalar function f which tends to 0 at the boundary of a space. Is it true that the volume integral of the gradient of f will also tend to 0?
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0answers
16 views

How to design Boundary condition for Euler equations (CFD)?

I'm developing on the calculation of the euler equations using the finte volume method. As you may know each cell is calculated by the incoming and outgoing flux. That means I need in a 1D System the ...
1
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2answers
53 views

Why isn't the volume formula for a cone $\pi r^2h$?

So I understand that the volume formula of a cone is: $\frac{1}{3}\pi r^2h$, and when I read about how to derive this formula, it makes sense to me. Funny thing is, I'm stuck on why it ISN'T $\pi ...
0
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2answers
22 views

Volume under surface

What is the volume under the surface $z = f(x,y) = x^4 + xy + y^3$ over the rectangle $R = [1,2] \times [0,2]$. I solved the double integral by integrating with respect to $x$ and then with respect ...
1
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1answer
39 views

Volume of sphere inside a cylinder

I have homework question: Calculate the volume of $\{(x,y,z) \mid x^2+y^2+z^2\le (2a)^2\} \cap \{(x,y,z) \mid (x-a)^2 + y^2 \le a^2\}$ in two ways: 1) Polar coordinates 2) Spherical coordinates ...
0
votes
1answer
29 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
1
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3answers
21 views

Volume of a rectangular prism's walls

Sorry if this is an obvious question... I have been trying to figure this out for a little while and come up with nothing... If I have a rectangular prism, say $5\times10\times12$ meters, it has a ...
11
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2answers
229 views

Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for $$ \text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)), $$ where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ ...
0
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1answer
29 views

I'm having trouble with this question on derivatives.

Carlos is blowing air into a spherical soap bubble at the rate of $7 \mathrm{cm}^3/ \mathrm{sec}$. How fast is the radius of the bubble changing when the radius is $11 \mathrm{cm}$? (Round your answer ...
1
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1answer
51 views

Volume of the cooking pot

A cooking pot has a spherical bottom, while the upper part is a truncated cone. Its vertical cross-section is shown in the figure.If the volume of food increases by 15% during cooking, what is the ...
2
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2answers
36 views

Volume of the solid of revolution generated when the parabola spins around the $x$ axis

Consider the bounded area by the straights $x=0,\;y=1$, and the parabola $y^2=4y-x$, calculate the volume of the solid of revolution generated when the parabola spins around the $x$ axis. I think ...
5
votes
2answers
213 views

Integral Calculus, right or wrong?

I have two questions, which I decided and my feedback does not match that of my book, but do not seem to be wrong. Calculate the volume of the solid obtained by rotating the region bounded by the ...
0
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2answers
43 views

How to find volume of a sphere

what is the volume of a ball that is 5.5 feet tall? I am trying to figure this out but i cannot figure it out with the information given.
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0answers
13 views

$H=\min(t_1,…,t_n)$ volume of a box in $R^n$

Please help me prove the following: n-Box is defined as $B=[a_1,b_1]\times[a_2,b_2]\times[a_3,b_3]\times...\times[a_n,b_n]$ Cartesian product of $n$ closed intervals, where $a_i$ and $b_i$ are all ...
0
votes
2answers
53 views

Double integral - Volume

I find it very difficult to solve this problem. I need help setting the integral up. I know that the first one is cone and the third one is paraboloid, but I can't define the limits. Some explanation ...
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1answer
20 views

Sketching and Finding Volume of a Container

I generally struggle with sketching graphs so some guidance would be appreciated - A container is made that encloses the volume defined by rotating the curve $x = z^{3/2}$ about the z axis, where ...
0
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1answer
26 views

Amount of spheres in a particular volume (Kepler Problem)

Sorry to ask, but I want to double check my math, and seeing in how I just learned a few things I do not trust my math(yet). I have an area I want to fill with 3mm size spheres. The cubic volume of ...
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1answer
15 views

Volume of an object obtained by rotating a region bounded by functions

The region bounded by $xy=1$ and $x=1, x=3, y=0$, rotated about $x=-1$. I figured out the fact that I have to use Cylindrical Shell method, since $$\int A_{outer} - A_{inner}$$ thing doesn't work in ...
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0answers
16 views

Find the volume of the figure generated by these relations.

Calculate the volume of a solid whose base is in a xy- plane and is bounded by the parabola y = 4 - x^2 and the line y = 3x while the top of the solid is in the plane z = x + 4. I'm able to somewhat ...
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0answers
25 views

Is there a diffeomorphism $f$ such that $V(B) < V(f(B)) <V(B)+V(B)^2$

Simple to understand calculus question that involved change of variable theorem in integration. Suppose $B$ is some open ball in $\mathbb R^n$, and $f: \mathbb R^n \to \mathbb R^n$ is a ...
2
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1answer
44 views

Calculate the volume of water in glass over time.

For A) I found that volume should be defined by But I got no idea what to do in b) and c)
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2answers
22 views

Volumes by cylindrical shells

Find the volume bound by $$xy=1;x=0;y=1;y=3; $$rotated about x-axis. In my attempt, I determined that the shell heigh will be 1/y and the shell radius will be y-1 Clearly these are wrong as the ...
1
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1answer
24 views

Volume cut sphere radius $= 4$ plane $z=2$

I have to calculate the upper volume of a sphere: $$x^{2}+y^{2}+z^{2}=16$$ cut by the plane $z = 2$ This is my approach: ...
1
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1answer
15 views

Negative volume in solids of revolution using the washer method

I need to find the volume of the solid formed by the area trapped between $\ y=x^2$ and y=√x, rotated about the y-axis. The two curves cross at y = 0 and y = 1 so they will be the upper/lower limits. ...
3
votes
2answers
41 views

Paradox of the trumpet shape

This is a question I had for long time now, when you rotate the function y=1/x, x>0 (say x and y both measure meters) about the x axes by 2pi you get a shape which has infinite surface area and finite ...
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0answers
13 views

Geometric figures inside a bounding box

Let us suppose that we have a bounding box of length $l$, width $w$, and breadth $b$. I wish to know the dimensions / characteristics of a tetrahedron cylinder cube pyramid inside this bounding ...
2
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4answers
57 views

Negative Volume using $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$

So, my textbook explains how to find the volume of a paralelpiped using $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$. Makes sense, basically. But, when I go to do problems some combinations ...
1
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0answers
18 views

Volume of overlap between two convex polyhedra

I have two convex polyhedra represented by triangle meshes. I can easily determine if they are in contact or not, but when they are in contact then I would like to determine the volume of their ...
1
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1answer
32 views

Fitting small cubes into a bigger container

This is a simple question that I have done on a test paper, but the given answer is driving me mad. The question essentially states that a person has a bunch of cubes of volume 3 cubic centimetres. ...
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5answers
44 views

A pyramid has a square base with sides of length 4. If the sides of the pyramid are equilateral triangles, what is the pyramid's volume?

A pyramid has a square base with sides of length 4. If the sides of the pyramid are equilateral triangles, what is the pyramid's volume? (A) 9.66 (B) 11.39 (C) 12.58 (D) 14.12 (E) 15.08 I know that ...
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0answers
17 views

Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
1
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0answers
32 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
1
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0answers
27 views

Volume of the set of two bodies

I have been given two bodys: $(x^2+y^2) < R^2(x^2-y^2)$ $x^2+y^2+z^2 < R^2$ Now I am supposed to calculate the volume of the set of the two bodies. I can see that the second body is a ...
0
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0answers
19 views

Flow with divergence-free generating vectorfield: Conservating volume

A function $g\colon M\to M$ is called to conserve volume if for any Jordan-measurable subset $J\subset M$ it is $\text{vol}(g^{-1}(J))=\text{vol}(J)$, Now I would like to show that for a ...
1
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2answers
25 views

Set up: Volume by Integration

Having difficulty setting up the equation. Find Volume bound by $$y = x; y = 0; x = 2; x = 4$$ and rotated about $x = 1$ The issue I'm having is that I get a final answer of $\frac{16\pi}{3}$ ... ...
2
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1answer
29 views

Find volume of revolution

Find the volume: $$x=y^2; x=1-y^2; \text{ rotated around }x=3$$ Use the disk/washer method. My inner radius is $3+y^2$ and outer radius is $3-y^2$ and the limits of integration are $\frac1{\sqrt2}$ ...