For questions related to volume.

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0answers
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Calculate rotational volumes

I need to calculate the volume from rotating f(x) around y=2x using Pappus–Guldinus theorem. For that I need to know the distance A. $$L = (f(x) - 2x) / 2$$ But how can I optain the distance A?
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1answer
19 views

finding volume of an n-dimensional pyramid numerically

In my experiment I need to compute hypervolume/area from a set of points, let's start with a base case -- Triangle: In this case, I have 3 points in a 2D space and they make a triangle, $p_1 = ...
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2answers
34 views

volumes solving for dx or dy

The only problem I have with this is knowing when you are solving for dx or dy. For example, this question which says find the volume of the solid created by rotating the region bounded by y = 2x-4, ...
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0answers
30 views

minimal volume of pyramid x0,y0,z0 [on hold]

I dont know how should i even start. I tried to think about something but get nothing. can someone help me please? Thanks
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1answer
25 views

volume of a triangular pyramid [on hold]

A point M0 with positive coordinates (x0, y0, z0) (x0>0, y0>0, z0>0) is given. A plane is drawn through this point. Find the minimal possible volume of a triangular pyramid whose faces lie in the ...
-3
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0answers
14 views

How to find volume using Riemann sum with expression $e^x + 3x^3 - x^2$? [on hold]

I'm desperate. It is for math class so please help if you know how to find volume using Riemann sum, and not double integral.
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0answers
15 views

Question about volume/area of cone and hemisphere [on hold]

Can any one answer this question? + i dont have any answers of it cz it came in my recent cie paper but still i am confused what could be the possible solution for this.
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1answer
35 views

Finding the largest box inscribed in the ellipsoid

Among all rectangular boxes inscribed in the ellipsoid: $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ How to find the one with the largest volume?
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0answers
32 views

Volume between a cone and a sphere

I am looking for the volume between a cone : $x^2 + y^2 = 2(95-z)^2$ and a sphere : $(x-100)^2 + (y-100)^2 + (z-150)^2= 125^2$. I want to determine the volume of their intersection. I tried to change ...
0
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2answers
25 views

finding volume of solid

Suppose that a solid is formed in such a way that each cross section perpendicular to the x-axis, for $0 \le x \le 1$, is a disk, a diameter of which goes from the x-axis out to the curve $y = ...
2
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2answers
28 views

Cylinder cut out of a sphere. (volume).

1.) A cylinder $Z=${$(x,y,z)\in R^3|x^2+y^2\leq \rho^2$}$~~$ ($0<\rho<R$) is cut out of a sphere $K\in R^3$ with radius $R>0$ centered around the origin. Find the volume of the rest ...
2
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1answer
25 views

Finding volume using washer method

I'm supposed to determine the volume of the region obtained by revolving the region lying below the graph of the given function and above the $x$-axis about the specified axis. The problem I'm given ...
2
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1answer
25 views

Calculating volume by shell integration

$y = \ln x$, region is delimited by $y = -1$, $y = 2$ and the $y$-axis, it rotates around the $y$-axis. It's quite simple to solve by using disk integration but I can't get it right with shell ...
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1answer
22 views

n-simplex volume and triangle.

For $n\in N$ let $\sum_n(1)$ be the standard-simplex. Let there be a point $b\in R^n$ and a basis {$a_1,...,a_n$} of $R^n$. The $n-simplex$ set up in this point b by the basis is the set ...
2
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2answers
70 views

Curve fitting the cross sectional area of a cake.

For my final Calculus project I have to find the area of a Bundt cake through the use of cross sectional areas. (Cakeulus) While most seniors in High School who run into this popular calculus project ...
0
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1answer
22 views

Finding volume of a revolution

I want to find the volume of the revolution that occurs when the region bounded by $y = x^2$ and $y = 1$ is revolved around the line $y=2$. The problem is that it is not solid and I cannot understand ...
0
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3answers
52 views

If the surface area of a box is 32 and its volume is doubled what is the new surface area? [closed]

Original surface area :32 Original volume: x New volume: 2x What is the new surface area? Please provide an explanation or show work, I don't know how to do it.
2
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1answer
20 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
3
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1answer
18 views

Volume of Convex Polytope with rational Entries

I have the following question: In this article Polytope volume computation it is stated that when considering a bounded convex polytope $P=\{x \mid Ax\le b\}$ with the matrix $A$ and the vector $b$ ...
0
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0answers
27 views

Using the general slicing method to find the volume of a semi-circle whose cross sections are squares.

In finding the volume of a solid, described below, I was close in finding the equation, but neglected a coefficient. Please see the question below. Use the general slicing method to find the volume ...
0
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1answer
44 views

Find the volume of the solid in $\Bbb R^3$

I need to find the volume of the solid in $\Bbb R^3$. It is bounded by the following: $y=x^2$, $x=y^2$, $z=x+y+21$ and $z=0$. I known that the volume is expressed as follows: $$\iiint 1 \, dV$$ I ...
1
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1answer
44 views

n-dimensional volume. Need some help.

For a>0 we define $\sum_n(a):=\big\{(x_1,...,x_n)\in R^n|x_1\geq 0,...,x_n\geq 0, \sum_{k=1}^nx_k\leq a\big\}$ 1.1.: Show that the n-dimensional volume $v_n(\sum_n(a))=\frac{a^n}{n!}$ 1.2.: ...
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2answers
29 views

Two approaches to the Volume of a Drilled Sphere (Self-Answered, open to other answers)

Find the ratio of the volume of a sphere of Radius $R$ with a hole through its vertical axis in the shape of a coaxial cylinder with radius $r$, to the original volume of the the sphere with raidus ...
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0answers
12 views

Interpretation of infinitesimal measure in Lebesgue integration

I have a little trouble understanding the notation of the infinitesimal measure in Lebesgue integration. For example, let's assume I want to compute an volume integral of a function $f: D \rightarrow ...
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2answers
40 views

<Finding the volume in a cylinder that is intersected by a plane

I have a homework question that goes as follows. Let $V = \{(x, y, z) : x^2 + y^2 \leq 4 \text{ and } 0 \leq z \leq 4\}$ be a cylinder and let $P$ be the plane through $(4,0,2), (0,4,2)$, and ...
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0answers
16 views

Divergence theorem for a box

I have to use the divergence theorem to calculate $$\int \int_C F.dS $$ for $$F = (x^2z^3, 2xyz^3,xz^4) $$ where S is the surface of the box with vertices at $(\pm1,\pm2,\pm3)$ with outward pointing ...
4
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2answers
112 views

How many ways are there to express a number as the product of groups of three of its factors?

Specifically, I am thinking of a cuboid with a given volume ($28\,000$) that has sides of integer length. For example, $20 \cdot 20 \cdot 70 = 28\,000$, but so do $10 \cdot 40 \cdot 70$ and $1 \cdot 1 ...
4
votes
1answer
157 views

Aproximating the volume of a sphere by dividing it into infinitesimal cubes

I have spent hours trying to solve it by substituting into two right triangles, asked multiple teachers but none can answer... This is not a homework, its just out of pure curiosity. I did the same ...
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0answers
24 views

Find the volume of the solid generated by revolving the region across the $y$-axis

Find the volume of the solid generated by revolving the region across the $y$-axis bounded by the graphs of the equations: $x=y^2, x=20y-y^2$, the line $x=102$. I set up an integral from $0$ to $10$ ...
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1answer
25 views

How to find the volume of a solid of revolution?

The region bounded by $x^2-y=0$ and $x+y=0$ is rotated around $y=1$ I solved for: $$y = x^2 $$ and $$y = -x$$ And I then also solved for x for both of those. I set up the integral like so: $$ V = ...
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1answer
18 views

volume using spherical coordinates

Let $$V = \{(x, y, z): x^2 + y^2 ≤ 4 , 0 ≤ z ≤ 4\}$$ be a cylinder and let $P$ be the plane through $(4, 0, 2), (0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I'm having ...
5
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1answer
55 views

Compare determinants of matrices with different dimensions

Reading about matrices and determinants I am wondering about the following concept: How valid is to compare the determinants of matrices with different dimensions? e.g. compare a determinant $D1$ ...
0
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1answer
47 views

How to calculate the volume of the solid described $x^2+ y^2+z^2 \le 9$ and $2 \le z \le \sqrt5$?

How to calculate the volume of the solid described $x^2+ y^2+z^2 \le 9$ and $2 \le z \le \sqrt5$? I try $x=2r \cos \phi$, $y=2r \sin \phi$, $z=z$, but but probably not the way to go
2
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0answers
31 views

Using Gauss's theorem to determine the volume

I'm currently stuck with this kind of problem: Let $S$ be the surface in $\mathbb{R}^3$ obtained by evolving the curve $x = \cos u$ and $z = \sin 2u$ where $-\frac{\pi}{2} \leq u \leq ...
0
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0answers
35 views

Cylindrical/ Spherical Coordinates Integral - Volume of Cone

"(b) Let $C$ be the solid cone with the boundary surfaces $x^2 +y^2 = z^2$ and $z = 0$. The density of the solid at point $(x, y,z)$ is $z$. Find the volume of the solid using the integrals in both ...
0
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2answers
62 views

Matlab Monte Carlo Method Volume Calculation

The question: use the Monte Carlo method to find the volume of intersection of, three cylinders, all of radius 3 units and infinite in length,with the axis of the first cylinder being the x-axis, the ...
0
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1answer
23 views

How can I find the volume of a solid bounded by these functions?

The region bounded by $x^2 - y = 0$ and $x+y=0$ is rotated around $y=0$ I've drawn a graph of each of these but I'm just not getting it conceptually. The biggest problem I have is knowing when to use ...
2
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2answers
36 views

How do I prove that the area of a sphere is the least possible area for a given volume?

Or, why do soap bubbles have the shape of a sphere and not that of unicorns? What's the math behind it?
15
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1answer
91 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...
2
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0answers
28 views

What is the volume inside $S$, which is the surface given by the level set $\{ (x,y,z): x^2 + xy + y^2 + z^2 =1 \}$?

The solution given uses a linear algebraic argument that doesn't seem very instructive -- and may not even be correct, I think. We notice from the equation, that the surface is a quadratic form, ...
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2answers
49 views

A hole of radius 1 is drilled to the centre of a ball of radius 2

A hole of radius $1$ is drilled into the centre of a sphere of radius $2$. Calculate the remaining volume. Could someone explain to me how to approach this. I don't need a specific answer. I know ...
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1answer
47 views

Calculating the volume of a cylinder.

Let $V = \{(x, y, z): x^2 + y^2 ≤ 4$ and $0 \le z \le 4\}$ be a cylinder and let $P$ be the plane through $(4, 0, 2)$, $(0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I ...
0
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1answer
12 views

Does a bounded countably infinite union of sets with volume have volume?

If $ A_1, A_2,...$ are sets with volume and $A= \cup_{i=1}^\infty A_i$ is a bounded set, must $A$ have volume? This was a homework problem that we went over in class, and if I remember correctly the ...
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1answer
46 views

Cavalieri’s Principle for calculating volume.

Let $B = \{(x, y, z): x^2 +y^2 +z^2 ≤ 4\}$ be the ball with radius $2$ in $\mathbb{R}^3$ and let $V$ be the region inside $B$ above the plane $z = 1$. Use Cavalieri’s Principle to compute the volume ...
2
votes
1answer
59 views

How to calculate the volume of a skip bin container knowing the height of the material inside

I need to know hot to calculate the volume of a skip bin (also known as a skip container or dumpster in some areas) with varying length and width. It seems like a isosceles trapezoid when you look at ...
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0answers
22 views

Difference between measure zero and volume zero?

I have the following definitions for a set to have measure zero and for a set to have volume, respectively: A set $A$ has measure zero if for any $\epsilon > 0$ there is a covering $\{S_i\}_{i \in ...
0
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1answer
43 views

Integration: Finding area, volume and arc length

I am new to integration, so please do not mark this question as "not enough research done" Here is the question (please open image in new tab to see it clearly) - I am getting stuck with the ...
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0answers
51 views

Volumes of solids (of rotation). Any real world applications?

Washer method, disc method, etc. In what areas or fields would someone make these calculations? For example, I think 3D printers do some sort of "slicing" algorithm in their CPU in order to print ...
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1answer
28 views

displacement of water

I just read the following puzzle on puzzling SE and it made me wonder, is there a proof which can be formulated to see if an object of any size and weight would cause more displacement inside the boat ...
5
votes
1answer
69 views

Volume of a hole through cylinder (from the side)

I need to calculate the volume of a circular hole in a cylinder and I've come across a problem. The problem is finding the "cap-volume", which is needed to complete the volume of the hole. I created a ...