Questions tagged [volume]

For questions related to volume, the amount of space that a substance or object occupies.

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Volume of the solid in the first octant bounded by the coordinate planes, a cylinder and a plane [closed]

Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder $x^2 + y^2 =4$ and the plane z + y = 3
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Calculate the volume of solid E bounded by the surfaces $z=4-x^2$, $z=4-y^2$ and $z=1$ [closed]

This question popped up in my calculus homework. We are doing multiple integrals.
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How to calculate Volume between surface and a tangent plane

I want to calculate the volume between $c:\ 4x^2+9y^2+z^2=36$ where $x>0,y>0,z>0$ and its tangent plane at the point $(x_0,y_0,z_0)$. To find perpendicular vector $\vec{a}$ to the plane let ...
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1answer
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^...
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1answer
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How can I assemble an equation from these guide points?

For context, I am programming a building system and I want to encourage players to build with larger blocks for a given volume of world space and discourage occupying the same space, but with multiple,...
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23 views

Weighted volume of simplices making up a cube not equal to volume of cube?

I am trying to implement an algorithm to calculate the volume of a polyhedron by dividing it into simplices with their apex as some arbitrary vertex of the polyhedron and then summing up the volume of ...
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1answer
70 views

Volume of a solid with specific cross sections

We can use definite integrals to find volume of a solid with specific cross sections on the interval. If the cross section is perpendicular to $x$-axis and its area is a function of $x$, say $A(x)$, ...
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Volume of unit nuclear norm ball [closed]

Let $M_{n\times m}$ denote the vector space of $n \times m$ matrices. The nuclear norm is defined by $$ \| M \|_* := \sum_{i=1}^{\min(m,n)} \sigma_i (M) $$ where $\sigma_i (M)$ are the singular values ...
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2answers
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Calculating Gabriel's Trumpet volume $\left(\frac{1}{z}\right)$

I want to calculate the volume of the solid obtained by rotating the function $1/z$ about the $z$-axis for $z>1$, that is $\pi$. But I want to apply that: \begin{equation*} V = 4\int \int_{\...
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1answer
36 views

Find volume between two surfaces [closed]

I need to find the volume between $$x=z^2+2y^2, \space 2z+4y+x=1,$$ but I can't imagine the body I have. I tried to draw this surfaces, but it seems to me that I need something else.
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Why do I not get volume units from my expression?

In my course we had to calculate the inner volume of a paraboloid described with $f(x, y) = x^2 + y^2$ by converting cartesian coordinates to polar coordinates. The formula I ended up with was $f(r, \...
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3answers
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volume cylinder [closed]

It's a silly question, but still: Is it possible to calculate the volume of a cylinder only knowing the area $A_1$? (I don't know what it's called. It equals $r \cdot h$.)
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Find the volume of the solid generated by revolving the shaded region about x-axis

Question:- Find the volume of the solid generated by revolving the shaded region about $x$-axis( The curve is $x=2y-y^3$). I think we will not get overlapping volumes due to symmetry of the shaded ...
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The Area Between Two Parametric Spiral Curves

Four points are given, $P(x,y,z)$, $P'(x,y,z+c)$, $Q(0,0,z)$ and $Q'(0,0,z+c)$. Knowing that $$x(t) = a\sin(4\pi t), \quad y(t) = a\cos(4\pi t), \quad z(t) = t$$ Where $a$ is a constant and $$0 < c ...
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1answer
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volume preserving version of Moser's theorem

There exists an well-known theorem of Moser : Thereom(Moser) Let $M$ be a compact oriented smooth manifold and $\alpha,\beta$ be volume forms whose total volumes are the same. Then, there exists a ...
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1answer
64 views

How to calculate the volume of this set using spherical coordinates?

Let $$K = \left\{(x, y, z) \in \mathbb R^3 ; \; x^2 + y^2 + z^2 < b^2, \, x^2 + y^2 > a^2 \right\}.$$ How to calculate the volume of $K$ using spherical coordinates? I know that the conditions ...
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Volume generated by revolving the region bounded by the curve $y=x^2+1$ and the line $y=-x+3$ about the $y$ axis

Question:- Find the volume of the solid of revolution generated by revolving the region bounded by the curve $y=x^2+1$ and the line $y=-x+3$ about the $y$- axis. The region on the right when revolved ...
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Maximize DIY Aquarium Volume Given Dimentions of 2 Scrap Pieces of Glass

I have 2 extra pieces of glass and was wondering what the maximum volume DIY aquarium I could make was. The first piece is 10.5 by 15+15/16. The second piece is 9 by 18. All measurements are in inches....
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A twisted Cavalieri's Theorem Problem

Question:- Find the volume of solid that lies between planes perpendicular to the axis of symmetry of cone and between $x=0$ to $x=12$. The cross section by planes perpendicular to the axis of ...
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1answer
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Why does changing integral bounds get me the wrong answer?

Full disclaimer, this is a homework question. While solving this question, I came upon the integral $$\int_{-r}^{r}\frac{b\tan^{-1}(\theta)}{2}\sqrt{r^2-x^2} dx$$ Proceeding with trig substitution I ...
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1answer
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Volume of a Parametric Surface

The cone pictured above has a radius of 1.5cm and a slant height of 3.5cm. Letting $\alpha=\arcsin\left(\frac{3}{7}\right)$ and $h=3.5\cos\alpha$, I have parametrized the cone in two ways: $r(t,u)=\...
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0answers
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What is the maximal volume of the torus?

We have a $10 \, {\rm m}^2$ metal plate and we want to construct a toroidal gas tank, as shown below. Assuming that all metal is used, what must the values of inner radius $a$ and outer radius $b$ be ...
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3answers
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Proof for volume of n-ball with radius 1

I'm trying to prove this formula from here, that the volume of a n-ball with radius 1 (let's call it $B_n$) is: $$\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}$$ However, I come to the wrong result and I ...
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Volume of unit sphere by iterative integrals

I want to calculate the volume of the unit sphere: $\{(x,y) \in\mathbb{R} \,|\, x^2+y^2 \leq 1\} = K$ I am supposed to use iterative integration. We have defined the volume as: $\int_\mathbb{R^2}\,\, ...
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Why does $V=\frac{1}{3}Bh$ hold for any pyramid? [duplicate]

For some personal math study I was looking at the derivation for the general pyramid volume formula ($V=\frac{1}{3}Bh$). I understand where this comes from in the case of a square-based pyramid, but I ...
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1answer
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Volume in spherical units

I'm working on Cavendish's original analysis of his experiment. Cavendish does not use equations or standard units. For instance, he states the weight of the lead weights by taking the weight of a ...
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3answers
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Triple integral of a sphere bounded by a region

The question is: Calculate the following integral $$I = \iiint\limits_D (x^2 + y^2 + z^2) \,dxdydz$$ where D is the region limited by $x+y+z\le2, x \ge 0, y \ge 0, z \ge 0$ My attempt was trying to ...
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1answer
51 views

Setting up the triple integral in $\mathrm {dz dy dx}$ order

Question:- Find the volume of the region bounded by $x+z=8 ,z=x ,y=8, z=y$ and $z=0$. The region looks like a prism with vertical base( $y=8$) on one side and slanted base( $z=y$) on other side. $$V=\...
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1answer
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Volume of Trapezoidal Prism

For a plot of land of 100 m × 80 m, the level is to be raised by spreading the earth from a stack of a rectangular base 10 m × 8 m with vertical section being a trapezium of height 2 m. The top of the ...
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4answers
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Find the volume of the solid bounded by $z^2=xy$; $x+y=a$; $x+y=b$ $(0<a<b)$ by applying variable substitution

Find the volume of the solid bounded by $z^2=xy$; $x+y=a$; $x+y=b$ $(0<a<b)$ by applying variable substitution. $xy>0$ and $x,y$ can't be negative at the same time so $x>0;y>0.$ Now, $a ...
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1answer
40 views

Volume of a container with unknown side and known surface area

We have to make a closed container. All side surfaces (walls, ceiling and bottom) must be rectangles and stand perpendicular to each other. One of the sides should be $3 \,m$ long. The surface area ...
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Find milk in cylindrical base on frustum

A container made of metal is in the shape of a frustum of a cone mounted on a hollow cylindrical base of the same metallic sheet. The diameter of the two circular ends of the container are 50 cm and ...
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2answers
311 views

Divide a ball of volume $\frac{e^2}{6}n$ into $n$ slices of equal height. What is the product of the volumes of the slices as $n\rightarrow\infty$?

Divide a ball of volume $\frac{e^2}{6}n$ into $n$ slices of equal height, as shown below with example $n=8$. What is the limit of the product of the volumes of the slices as $n\rightarrow\infty$? (If ...
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0answers
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Is the following cartoon mathematically correct?

I saw the following cartoon about probability and statistics the other day: The joke in the above picture being that "the top 1%" (i.e. the integral of the gaussian probability distribution ...
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Calculus proof for volume of a general pyramid

I want to use calculus to generalize the volume of a square pyramid to a pyramid with any base. Is the proof below correct? Are there conditions/restrictions Introduce a $z$ axis perpendicular to the ...
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1answer
87 views

Volume of a body bounded by a surface.

I want to find a volume of a body, bounded by: $$(x^2+y^2+z^2)^2 = az(x^2+y^2)$$ for some $a > 0$. As I understood I am supposed to say that for аor a fixed value of z, it looks like a circle, ...
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1answer
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Volume of a Torus Segment

I am calling a segment analogous to a circle's, that is a splitting a torus in two by a chord. The resulting piece has an ellipse formed by the chord. The chords distance from the torus centre lies ...
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Finding region bounded by $z=1-x^2,\ z=x^2-1,\ y+z=1,$ and $y=0$ returns negative volume

Find region bounded by $z=1-x^2,\ z=x^2-1,\ y+z=1,$ and $y=0$ This question seemed simple to me what I did was the following: when plotting the x-z graph I find two parabolas creating a leaf-shaped ...
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0answers
75 views

Upper-bound on volume of polytope inscribed in the sphere

To my great surprise, I was unable to find any general reference on the volume of (symmetric) polytopes inscribed in, say, the unit sphere. Let $P = absconv(v_1,\dots,v_k) $ where the $v_i$'s are unit ...
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1answer
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Bounding Circles

If I have a set of points P in 2D I can compute a minimal bounding circle encapsulating all points in the set. Different methods either approximate the optimal solution or search through all the ...
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1answer
36 views

Volume bounded by a plane and parabolic cylinder

Find the volume of the region in the first octant bounded by $z=x^2+2$, $x=y=z=0$, and $x=2-y$. Let the order of integration be $dzdxdy$. Here's what I did: $\int_0^2\int_0^{2-y}\int_2^{x^2+2} dzdxdy$....
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2answers
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Find the volume between the sphere $x ^2 + y^2 + z^2 = 4$ and the plane $z = 1$

Suppose $y \geq 3$. I want to compute the volume between the sphere $x ^2 + (y − 2)^2 + z^2 = 4$ and the plane $y = 3$. So I move left the sphere and and the plan, and rotate it counterclockwise. I ...
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0answers
27 views

What is the content of an m-parallelotope in n dimensions?

How do I calculate the m-dimensional content of a parallelotope given by m vectors which are all n-dimensional (m<n)? For example 3 vectors spanning a parallelepiped in high dimensional space. I ...
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1answer
25 views

Parametric Volume, volume analogue of parametric surface?

Is it possible to have a parametric volume? Is it possible to have a parametric volume in $\mathbb{R}^3$, $V:\mathbb{R}^3\rightarrow\mathbb{R}^3$, where $V$ is some function $V(x,y,z) = A(x,y,z)\hat{i}...
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0answers
79 views

Find the volume of a solid bounded by planes x=0, y=0, z=0 and 2x + 3y + z = 6

Given question: Find the volume of a solid bounded by planes x=0, y=0, z=0 and 2x + 3y + z = 6 My doubts: Apparently, this is a tetrahedron so we can find the volume by double integration and setting ...
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1answer
28 views

How do I use the shell method to calculate the volume of this region between two solids of revolution?

My question is how do I calculate the volume of the region between $f(x) = (2x+1)\sqrt{x^2 + x}\,$, $\,g(x) = x^2\,$, and $x = 1$ using the Shell Method? Here are the 3D visuals I graphed using ...
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1answer
56 views

Volume between two cones with intersecting axes

I have two cones with given vertex coordinates, axis direction and aperture angle. I know that the axes of the two cones are intersecting in a point (see the figure attached), which coordinates are ...
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1answer
40 views

Finding area between a curve and a line using double integrals?

We are asked to find the area between the circle $x^2 + y^2 = 4$ and the line $x + y = 2$ lying in the first quadrant. One method is to find the area by integrating $2-x$ from $0$ to $2$ and ...
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3answers
20 views

Volume around $x$ axis using shell method

I know the volume around $x$ axis of the region $$\mathcal{R}=\{(x,y)\in\mathbb{R}^{2}\mid 0\leq x\leq 1, y\leq e^{-x}, y\leq e^{x-1}, y\geq0\}$$ is (using the disk method) $$V=\pi\left\{\int_{0}^{1/2}...
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2answers
48 views

Calculating the volume of the solid defined by $x^2+y^2+z^2 \leq 2a^2$ and $z \leq \frac{x^2+y^2}{a}$, with $a>0$

I have to calculate the volume of solid geometry for $a>0$ $$T=\{(x,y,z)\in \mathbb{R}; x^2+y^2+z^2 \leq 2a^2; z \leq \frac{x^2+y^2}{a}\}$$ I know that first formula is inside of sphere with ...

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