Skip to main content

Questions tagged [volume]

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

Filter by
Sorted by
Tagged with
-2 votes
0 answers
25 views

Find the volume of the region intercepted by the plane $x+y+z = \frac{3a}{4}$ and the coordinate axes [closed]

Find the volume of the region intercepted by the plane $x+y+z = \frac{3a}{4}$ and the coordinate axes. Try to obtain the answer without the use of triple integrals, as I do not know how they work.
BeastlyFalcon's user avatar
0 votes
0 answers
14 views

Let $\Omega$ be the region in the first quadrant, enclosed by $y = 0$, $y = 3x$ and $y = -x^2 + 4$. Find the volume of the solid generated...

Could you help me to see if my analysis is good or wrong? Let $\Omega$ be the region in the first quadrant, enclosed by $y = 0$, $y = 3x$ and $y = -x^2 + 4$. Find the volume of the solid generated by ...
bdvg2302's user avatar
  • 1,159
0 votes
1 answer
41 views

Volume around $y$ axis

To find the volume of the solid of revolution around $y$ bounded by $$y=x^2,\quad y=x-2$$ and the lines $y=0$ and $y=1$, I did as follows: since the region is Then, the volume is: $$2\pi\cdot\left(\...
mvfs314's user avatar
  • 2,075
0 votes
0 answers
19 views

a cool optimization problem involving cubes surfaces and volumes

Consider a codimension one surface of revolution $S$ and an embedding $e:S \hookrightarrow X^n$ for $X^n=[0,1]^n$ with points $p,q$ elements of $\partial X^n$ where $\partial X^n=X^n-(0,1)^n$ for $\...
John Zimmerman's user avatar
0 votes
1 answer
47 views

How to find the volume of a tetrahedron that isn't aligned with flat planes with triple integrals

Hi I'm having a lot of trouble solving this math problem, I have been working on it for about 3 hours (please help). I'm learning about triple integrals and I have no idea how to solve the volume of a ...
chqlr's user avatar
  • 3
0 votes
1 answer
35 views

How is formula of the volume of trirectangular tetrahedron $\frac{A\times B\times C}{6}$?

I am just trying to understand where I am wrong. Let a cuboid be constructed with sides $A \hat{i},B \hat{j}, C \hat{k} $ Then the volume of the cuboid is $ A\times B\times C$ Trirectangular ...
sai prabhav's user avatar
1 vote
1 answer
30 views

Triple Integrals for volume

I got stuck with this problem because I don't know how to find the values por $p$. I also got $0$ to $\pi/2$ for $\theta$ and $0$ to $\pi/4$ for $\phi$ Let $S$ be the solid in the first octant inside ...
Ionixorca's user avatar
0 votes
1 answer
49 views

Shell and disc method applied to a specific integral [closed]

After watching Khan Academy, Org Chem tutor and a few others, my understanding is this: whether you solve in terms of $x$ and $y$ entirely depends on which axis you work with. For shell - if you are ...
Anish Shah's user avatar
0 votes
1 answer
31 views

How do I find the volume of this body?

I need help with finding the volume of the body given by $B = \{(x,y,z) \in D \colon 0\leq xyz \leq t^3\}$ Where $D \subset \mathbb{R}$ is given by $0 \leq x \leq t, 0 \leq y \leq t, 0 \leq z \leq t$. ...
nazorated's user avatar
1 vote
0 answers
12 views

Volume of Solid of Revolution Referred to by a Graph

I need to refer to the figure and find the volume generated by rotating the given region about the specified line. a is equal to 2, b is equal to 3, and we are rotating the region R1 about OC. enter ...
Blossompot's user avatar
2 votes
4 answers
361 views

A parallelepiped has a volume of $216 \text{ cm}^3$ and the total area $216 \text{ cm}^2$. Prove that the parallelepiped is a cube.

The problem A parallelepiped has a volume of $216 \text{ cm}^3$ and the total area $216 \text{ cm}^2$. Prove that the parallelepiped is a cube. My question I don't understand if we are talking about a ...
IONELA BUCIU's user avatar
0 votes
0 answers
43 views

Calculating the volume of a body given by $g(x,y,z) = xyz$ using triple integrals

I need help calculating the volume of the region/body/solid given by $D = \{(x,y,z) \colon 0 \leq xyz \leq 8 \text{ and } 0 \leq x \leq 2, 0 \leq y \leq 2, 0 \leq z \leq 2\}$, I am supposed to do it ...
nazorated's user avatar
3 votes
2 answers
224 views

Calculating the Volume with three given expression

How can I find the volume of the region defined by the expressions $z+x+y=180$, $40≤x≤100$ and $20≤y≤80$? I tried dividing this volume into familiar shapes (rectangular prism and triangular prism) but ...
Briston's user avatar
  • 192
0 votes
1 answer
51 views

How to calculate the Volume of an Elliptic Truncated Cone?

I have been attempting to find the volume of an Elliptic truncated cone by dividing it into cross-sections of elliptical cylinders and then stacking them up. I got the idea from the integration of the ...
Artimieus's user avatar
3 votes
2 answers
106 views

why cant i integrate the derivative of a cylinders volume to find it again?

For a cylinder, the volume formula is given by $V = \pi r^2 H $. To find the differential ( dV ), we differentiate ( V ) with respect to ( r ) and ( H ), yielding $\left( dV = 2 \pi r H dr + \pi r^2 ...
Youssef Badr's user avatar
0 votes
0 answers
33 views

A parallelepiped is formed using three non collinear vectors whose magnitudes are 1, 2, 3. Angle between any of the vector with....

A parallelepiped is formed using three non collinear vectors whose magnitudes are 1, 2, 3. Angle between any of the vector with normal of the plane determined by other two is π/3. Then the ratio of ...
Maths lover's user avatar
1 vote
0 answers
84 views

Volume integration spherical cap cut by plane

When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical ...
Klavier's user avatar
  • 11
0 votes
1 answer
34 views

Volume below the cone $z=2\sqrt{x^2+y^2}$ for $x^2+y^2\leq4$

For $x^2+y^2=4$, $$z=2\sqrt{4}\Rightarrow z=4$$ Since the radius of the basis is $2$, then the volume of the cone is $$V=\frac{\pi\cdot2^2\cdot4}{3}\Rightarrow V=\frac{16\pi}{3}$$ However, using ...
mvfs314's user avatar
  • 2,075
0 votes
1 answer
85 views

How is the volume not double counted in the shell method for volume of revolution?

If the radius of every shell $k$ for where k is the $k^{th}$ subinterval is,$\left(a + (k-\tfrac{1}{2}) \frac{b-a}{n}\right)$, you have a increasing radii sequence $(a+\dfrac{(b-a)}{2n}),(a+\dfrac{3(b-...
jkj's user avatar
  • 3
0 votes
1 answer
64 views

How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted?

Why is the shell method not $$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}...
user avatar
0 votes
1 answer
35 views

Finding the number of grams of gold in a solid sphere?

I am trying to solve a question from Leaving Cert. Ordinary Level Maths 2018 Paper 2. A solid sphere is made of gold. It has a volume of $0.113$cm$^3$. Each cm$^3$ of pure gold weighs $19.3$ grams. I ...
AlanaCut's user avatar
1 vote
1 answer
34 views

Why did the author need to extend $\alpha$ to a neighborhood of $Q$? ("Analysis on Manifolds" by James R. Munkres.)

I am reading "Analysis on Manifolds" by James R. Munkres. Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. ...
佐武五郎's user avatar
0 votes
1 answer
31 views

Volume of Same Region Yields Different Values

This question comes from a previous question on this site. There is the integral $$\int_{-1}^{1}\int_{0}^{\sqrt{1 - x^2}}\int_{0}^{\frac{y}{2}}f(x, y, z) \, dz \, dy \, dx.$$ Our job is to reorder the ...
Bored Comedy's user avatar
0 votes
1 answer
89 views

The method of calculating the volume of a parallelepiped halved along its entire length in a multi-dimensional space

Consider a right-angled triangle as shown below, where vertex $O$ is the right angle. If we consider the midpoint of the two legs and draw a line passing through it, we have proved that the area of ...
M a m a D's user avatar
  • 433
5 votes
1 answer
64 views

Find surface which generated by revolving a line in $\mathbb{R}^3$

Problem : Let $l$ be a line which passes two points : $(1,0,0), (1,1,1)$. And $S $ be a surface which generated by revolving line $l$ around $z$-axis. Find a volume enclosed by surface $S$ and two ...
bFur4list's user avatar
  • 2,751
0 votes
0 answers
58 views

how to integrate in the diagonal?

Imagine we have a function f(x,y). For simplicity, let's say: $$f(x, y) = \begin{cases} c & \text{if } 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ With ...
Oscar Flores's user avatar
-1 votes
2 answers
87 views

Which calculus method should I use for this investigation? What does it mean choose a suitable length for the cross section?

Your task is to determine which shape maximises the volume of water that can be held by a gutter of a fixed length of material. choose a suitable fixed length for the cross section and explore ...
math student's user avatar
2 votes
1 answer
67 views

Limits of triple integral over a tetrahedron.

The tetrahedron has vertices $O(0,0,0); A(0,0,2); B(0,2,0); C(1,0,0)$ I was thinking that the plane $ABC$ has equation $2x+y+z=2$ since: $\vec{BA}= \langle 0,-2,2 \rangle$ and $\vec{CA}= \langle -1,0,...
Badis Labbedi's user avatar
0 votes
1 answer
135 views

Volume of water in a tilted paraboloidal bowl

Suppose that you initially have a container which has the shape of a paraboloid with equation $ z = a x^2 + b y^2 $ where $ 0 \le z \le h $ Now you tilt this paraboloid by rotating it about any point (...
i don't know what i am doing's user avatar
0 votes
0 answers
46 views

I proved any linear subspace of $\mathbb{R}^n$ is closed in $\mathbb{R}^n$ to prove $\overline{T}$ has measure zero. (Munkres "Analysis on Manifolds")

I am reading "Analysis on Manifolds" by James R. Munkres. Theorem 20.1. Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation $h(x)=A\cdot x$. ...
佐武五郎's user avatar
1 vote
0 answers
79 views

Orientability issue when classifying symplectic structures on compact orientable surfaces

I have seen this post, where they identify symplectic structures on a compact orientable surface with $\mathbb R\smallsetminus0$, which I understand except for one issue regarding the orientation on $\...
Chris's user avatar
  • 3,177
2 votes
1 answer
30 views

What is the relationship between radius percentage and volume percentage for a layered sphere?

I'm attempting to form interior structures of planets. If I know the volume percentage of each layer of the planet, how would I determine the outer radius of each layer? Let's assume the planet is ...
ArceusLord13's user avatar
3 votes
2 answers
206 views

Calculate the volume of the tetrahedron...

The question Let $ABXY$ be a tetrahedron such that the triangles $ABX$ and $ABY$ are isosceles right angle in $A$ with legs of $2$ cm. Calculate the volume of the tetrahedron, knowing that there is a ...
IONELA BUCIU's user avatar
0 votes
0 answers
46 views

The intersection of $ n $ cylinders in $ 3$-dimensional space

I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set the radius $ = 1 $ the volumes are respectively $ V_2 ...
user967210's user avatar
0 votes
0 answers
36 views

Modeling of a conical base tank

Consider the following diagram of a empty tank being filled with a liquid Considering constant and equal denstity, it's total mass balance is $$\frac{dV_1}{dt}=q_e-q_s$$ But I don't know how to model ...
BadEnglishSorry's user avatar
0 votes
0 answers
24 views

higher dimensional analogue of volume difference of 2 three-dimensional cubes

In the illustration a way of expressing the difference in volume of 2 different sized 3-dimensional cubes is given: The volume of a black cube of size b is diminished by the volume of a red cube with ...
plopper's user avatar
3 votes
2 answers
127 views

Calculate the double integral: $\iint \frac{x^2}{x^2 + y^2}dxdy$ over the area $T = \{ (x, y) \in R^2 \mid 0 \le y \le x, \frac{1}{2} \le x \le 1\}$

I first started by setting the bounds inside the integral as follows: $$ T = \{ (x, y) \in R^2 \vert 0 \le y \le x, \frac{1}{2} \le x \le 1\}$$ $$\int_\frac{1}{2}^1 \int_0^x \frac{x^2}{x^2 + y^2}dydx$$...
Mohamed Boualem's user avatar
0 votes
2 answers
47 views

Volume - Calculus

I am having trouble with this question: Find a formula for the volume of the solid obtained by rotating the area between the curve $y = \smash{\frac{1}{\sqrt{x}}}$ and the $x$-axis between $x=a$ and $...
MSM's user avatar
  • 25
0 votes
3 answers
148 views

Stumped on $\int_{-1}^1\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} \int_{x^2+y^2}^{2-x^2-y^2} dz\;dy\;dx$

I cannot express the following integral in spherical coordinates. It as though I am finding the volume between the solids. Any help will be appreciated. $$\int_{-1}^1\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2}...
Jason Broadway's user avatar
1 vote
2 answers
156 views

Why can't I integrate with respect to x when using the shell method on a rotation about the x-axis?

The example I was given is find the volume of the solid obtained by rotating the region enclosed by $y=\sqrt{64-x^2}$ and $y=6$ about the x-axis. I first tried to integrate half of the region ...
Arjun Chopra 's user avatar
1 vote
1 answer
50 views

Volume of a solid generated by revolving a region about vertical line

Given the region bounded by these functions: $y = x, y = 0, y = 7$; how do I find the volume of that region revolving around the vertical line of $x = 8$? I've had trouble finding the answer to this ...
Ch1cken's user avatar
  • 13
1 vote
0 answers
38 views

Volume of a solid with an elliptical base using an isosoceles right triangle

I'm currently in the chair donating plasma so I'm sorry for lack of pictures or misspellings or anything. I'm trying to find the volume of a solid with an elliptical base (x^2)/49 + (y^2) = 1 using ...
karla's user avatar
  • 11
5 votes
3 answers
145 views

What is the volume of this shape, and how can I calculate it?

I have this shape with $7$ vertexes as specified. You start with a unit square, then at one back edge, two vertexes go directly above the bottom two with a height of $h$. Then a special final vertex ...
DrZ214's user avatar
  • 1,411
2 votes
3 answers
73 views

How do I express the region bounded by $z=x^2, z+y=1, z-y=1$ as a z-simple triple integral?

I've been able to figure out the y-simple and x-simple triple integrals, but I'm having trouble with the z-simple one. I split the region in half over the x-axis, from what I can see, the projection ...
Jabinski's user avatar
4 votes
1 answer
200 views

Calculate the distance between $AC$ and $BD$ in the regular tetrahedron $ABCD$.

Question Consider the regular tetrahedron $ABCD$. The points $M$ and $P$ are the means of the segments $BC$ and $AM$, respectively. On the edge $BD$ take the point $K$ so that $BK = 3\times KD$ ...
IONELA BUCIU's user avatar
1 vote
0 answers
69 views

Why I am getting a different answers when i am solving from 2d and 3d perspective??

This a very simple question from surface area and volume so the question is: A cube of maximum possible size is cut from a hemisphere of radius sqrt 6 cm. One identical cube is placed next to this ...
Surya Narayan adhikary's user avatar
1 vote
1 answer
80 views

Evaluating triple integral under linear transformation

Given linear transformation: $$T: \mathbb R^3 \to \mathbb R^3, T(x,y,x)= (3y+4z, 2x-3z, x+3y) $$ We need to evaluate the triple integral : $$\int \int \int_ {T(C)} (2x+y-2z) dx dy dz $$ where $C= \{{(...
S.S's user avatar
  • 1,229
0 votes
1 answer
50 views

Dimension of maximum volumed unit ball

Let $V_n=V(B^n)$ be the volume of the $n$-dimensional unit ball $B^n$. By cross-sectioning $B^n$ along $x_n$-axis, $-1\leq x_n\leq 1$ and by means of similarity of hyper disks we have $$V_n=2\int_0^1(\...
Bob Dobbs's user avatar
  • 11.3k
0 votes
1 answer
68 views

What is the term for a three dimensional region of space - as we have "surface" and "curve" for lower dimensions?

What is the term for a three dimensional region of space - as we have surface and curve for two and one dimensions respectively? I've sometimes seen this called a volume, although I find that ...
SRobertJames's user avatar
  • 4,410
1 vote
0 answers
58 views

Volume of the surface bounded by $x=\sin(y+\frac{\pi}{2})+2, -2\le{y}\le2, x\ge0$ rotated about the $x$-axis [closed]

I'm trying to solve this question on an assignment and I'm getting confused, especially since it's rotating about the x-axis and the question says disk method. The formula for the disk method is $V = \...
godhelpusall's user avatar

1
2 3 4 5
65