For questions related to volume.

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21 views

Complex Solids of Revolution

I know that to compute a solid of revolution of a function $f(x)$ rotated around the $y$-axis, one method we can use is the "shell" method. For example, $f(x)=1/4x^2\in [2,4]$, rotated around the ...
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2answers
27 views

Volume and Average Height

How do you calculate the average number of floors of buildings across a city block when all the buildings are varying heights. Followings in a listing of what the area looks like and the number of ...
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1answer
17 views

Find volume of cube with the help of eqn of plane

The volume of cube whose two faces lie on the plane 6x-3y+2z+1=0 and 6x-3y+2z+4=0?
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1answer
22 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
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1answer
31 views

Enlargement of area and perimeter in a rotation body

Let $f: [0,1] \to \mathbb{R}$ a continuous, differentiable function with $f \ge 0$. Rotate the graph of $f$ around the x-axis. Define this rotation body in $\mathbb{R}^3$ with $A$ and the area in ...
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0answers
6 views

On volume of arithmetic subgroups

I deal a lot with volumes of arithmetic subgroups, mainly in $SL_2(\mathbf{Z)}$. But I remain not at ease with them, making rough explicit calculations case by case instead of having a general method. ...
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2answers
45 views

Measuring water in a pool [on hold]

The edge of a swimming pool is composed of two equal arcs as in the picture below. The pool is 1.8 meters deep over all, and the pool is filled with water to 20 cm from the top. How many gallons of ...
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4answers
46 views

A cube and a sphere have equal volume. What is the ratio of their surface areas?

The answer is supposed to be $$ \sqrt[3]{6} : \sqrt[3]{\pi} $$ Since $$ \ a^3 = \frac{4}{3} \pi r^3 $$ I have expressed it as: $$ \ a = \sqrt[3]{ \frac{4}{3} \pi r^3} $$ and, $$ \ 6 \left( ...
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2answers
55 views

Find The Volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$

Find the volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$. $x$ and $y$ range from $0$ to $2$. $$\int_0^2 \int_0^2 y-3 \,dy\,dx $$ is correct?
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0answers
23 views

Conceptual Understanding of Rate / Volume Analysis for Balance Sheet changes

Not sure if this is the right place to ask this but I searched and didn't find this question already asked. I am having a lot of trouble conceptually understanding the formulas behind a rate / volume ...
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0answers
33 views

Spherical, polar coordinates, volume of set.

Find the volume: $$\{(x,y,z)\mid x^2+y^2 \leq (z-1)^2 \leq 4-\frac{x^2}{2} - 2y^2, z\geq 1 \}$$ I've got the intersection of the following two basically: \begin{align} 1. & & & (z-1)^2 ...
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1answer
35 views

Computing volume of concave polyhedron

I have a circular grid with points uniformly distributed throughout it. See this: Each point has some nonnegative height assigned to it (i.e. height can be 0 on up). I'm trying to accurately ...
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1answer
24 views

Finding Volume of Revolution with Multivariate Calculus

For some function $f\left(x\right)$ it is possible to rotate it along the x-axis and find the area using $$\intop_{a}^{b}\pi\left(f\left(x\right)\right)^{2}dx$$ I'm curious how to do this with ...
2
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1answer
28 views

Calculus II: Find the Volume (Shell-Method)

Find the volume of the shape created when rotating the region(s) bounded by $y=\sqrt{x+1}, y=0, x=0, x=1$, about the x-axis. I know this is a rudimentary question. My issue is that I tried to test ...
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2answers
29 views

I have this seemingly simple volume of a solid of revolution, but the limits and function are unknown.

How can I possibly find the numerical area of the region without knowing the function itself or the limits?
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1answer
34 views

Volume of irregular polyhedron [closed]

Let H denotes a polyhedron with its base being a triangle PQR with PQ = QR = 1/2 unit and angle Q = 90 degree . The cover ( upper layer ) of the polyhedron is determined by the equation : z= 1/2 x - ...
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1answer
38 views

Triple integral vs double integral to find volume of an object

Is it possible to find the volume of an object bounded by two surfaces in both of these two ways?: -a triple integral of 1 dV (I know this works) -a double integral of the top surface - bottom ...
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0answers
34 views

Find the volume(solid) , transform rectangular function to polar function

if want find volume of this problem Under $ f(x)=x^2+y^2-4 $ and inside $ x^2 + y^2=9$ in plane $z=0$ Can I use this integration in polar functions? $$\int_0^{2\pi} \int_2^{3} (r^2 - 4) r ...
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0answers
13 views

Help finding boundries for integration

I'm trying to find the volume of the solid bounded by $y=x^3$, $y=2-x$ I have $0<=y<=1$ and $y^(1/3)<=x<=2-y$. Is there something wrong with this? I was also trying to find the volume of ...
0
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1answer
13 views

Unable to calculate integral for cross-section volume

I am trying to determine the volume of a solid by cross-sections, and I am having trouble determining the integral I should use to calculate it. Graph in question The base is bounded from $x = ...
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1answer
31 views

What is the volume of a smartie? [closed]

According to wikipedia: Smarties are oblate spheroids with a minor axis of about 5 mm (0.2 in) and a major axis of about 12 mm (0.5 in). What is the volume of one smartie?
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0answers
26 views

Finding volume of an F-22 raptor

I'm trying to find the volume of an f-22 using calculus. Having a hard time know where to begin, I've started to partition parts of diagrams in a b,w,h format, and then I would multiply the sums of ...
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0answers
9 views

Half Cylinder Tank Integration Method?

I am trying to get a semi-circle cylinder tank, and I was wondering how I would set up an integral for that.∫ A semi-circle's equation, or slice, would simply be: π√(10² - x²)² Δx (set radius to 10 ...
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0answers
26 views

Volume of a polar solid of known cross section

I am trying to generate a solid of known cross sections to be semi-circles for the cardioid $r = 1 + \sin\theta$. I figured that I would need to calculate the volume of individual wedges and sum all ...
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2answers
38 views

What are some commonly used containers that have peculiar exact volumes?

I am doing a project in calculus in which I must calculate the volume of some kind of container. I didn't wish to choose a boring object. So I wanted to examine the options from the extensive ...
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1answer
29 views

Volume and Triple Integrals

When finding the volume of a solid using triple integrals, is the "integral" always $\int \int \int $ $dV$, regardless of what the "z function" is? For instance, for the problem: Find the volume of ...
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0answers
14 views

Why calculating the volume of Birkhoff polytope is complicated?

It is known that, Calculating the volume of Birkhoff polytope in higher dimension is still open. I am not very good on it, trying to understand, why it is complicated? It would be really great if ...
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0answers
44 views

How to calculate the volume of an irregular thing?

I having an external math class that asked the class to calculate the volume of a pencil. How can I do it?I know about the cup trick, but how? I forgot it.
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1answer
15 views

Calculating interesting volume (Also general multiple variable function equation)

First off, What does $f(x,y,z) = 1$ mean? versus $f(x,y,z) = x$. I have trouble with this as I am not sure choosing an initial $x$ factors into the function. Similarly with choosing a corresponding ...
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4answers
103 views

Finding the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. Cylindrical coordinates.

I would like the answer to preferably be done using either using a surface integral, or an integral with substitutions. But anything other than this is alright, if nothing else exists. I have to find ...
3
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1answer
42 views

Volume Contraction

I need to determine if this system exhibits volume contraction: $\dot x =yz-x-x^3$ $\dot y =xz-y-y^3$ $\dot z =xy-z-z^3$ My approach is to just calculate the divergence of the vector field F: ...
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0answers
35 views

Volume of paraboloid that is cut with plane

How to calculate the volume of the paraboloid : x^2 + y^2 = z that is cut with x + y + z = 5 plane. Please give several methods if you can. Thank you very much for answers.
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2answers
38 views

$2x^{1/4}$ rotated around $y = 2x$

This is the question: find the volume created by rotating $2 x^{1/4}$ around $y=2x$. I was able to define the distance between the two lines as $y/2 - (y/2 )^4$. However, I can't find the radius that ...
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0answers
26 views

Using integration to find volume of a parabolic prism.

Suppose I have a solid with its profile being a parabola given by the equation $y = f(x)$. Its depth at any point is given by $g(x)$ and its width is a constant $k$. This essentially results in a ...
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0answers
113 views

Water filling problem in Blocks - Algebra Question

Consider a rectangular plot comprising $n\times m$ square cells on which $nm$ cement blocks of various heights are stored, one block per cell. The base of each block covers one cell completely, and ...
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1answer
33 views

Ice in Skating Rink Weighs [closed]

How would you estimate how much all the ice in a skating rink weighs? Use variables like D for depth, L for length, and such. I am just stumped because if you get the volume, you turn it into mass, ...
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0answers
16 views

Volume Zero of Not Continuous Function

Show that a bounded real-valued function f on a closed interval $I$ of $E^n$ is integrable on $I$ if and only if the set of points of $I$ at which $f$ is not continuous is the union of a sequence of ...
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1answer
25 views

Find the length of the longest diagonal of the bo

The total length of all $12$ sides of a rectangular box is $60$. The surface area of the box is given to be $56$. Find $(i)$ the length of the longest diagonal of the box $(ii)$ the volume of the box ...
2
votes
1answer
87 views

Volume from equation $(x ^2+ y ^2 + z ^2 ) ^2 = xyz$

How can you calculate the volume of the shape represented by the following equation: $$(x ^2+ y ^2 + z ^2 ) ^2 = xyz$$ I tried converting it to polar form (so $r = ...
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2answers
48 views

How would the volume of a frustum with irregular polygon area be calculated?

I want to calculate the volume of this shape, it's basically a frustum with an irregular polygon base. The bottom area $A_1$, the height of the frustum shape $h$,the sideways distance between $A_1$ ...
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0answers
24 views

Estimating the volume of a beerglass

Most who have been to college recognizes the red glasses used for beerpong and alcohol consumption. My question is about why one method is better at estimating the volume and why. The exact volume ...
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0answers
41 views

Volume enclosed by Implicit Surface

I am trying to calculate the Volume that's enlcosed by the surface: $$(x^2 + y^2 + z^2)^2 = xyz$$ The following is what i tried. I rewrote it in spherical coordinates where $x=r \cdot \sin\vartheta ...
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0answers
9 views

Help in understanding limits of triple integrals in cylindrical coordinates

Question is: Use a triple integral to find the volume of the solid enclosed by the paraboloids $$ y = x^2 + z^2 $$ and $$ y = 8 − x^2 − z^2 $$. Their intersection: y=4 Integral giving correct ...
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0answers
22 views

Find the volume of the solid that lies under a paraboloid and above xy-plane.

Find the volume of the solid that lies under the paraboloid $z = x^2+ y^2$, above the xy-plane, and outside the cylinder $(x – 1)^2 + y^2= 1$. I would love to post some work that I've done but I ...
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1answer
35 views

Find the volume obtained by totating the area formed by $y=x$ and $y=\sqrt{x}$ about $y=1$

The questions asks us to find the volume of solid formed when the area between $y=x$ and $y=\sqrt{x}$ is rotated about the line $y=1$. I understand that a cone is formed. Now, to find the volume, I ...
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0answers
28 views

Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: ...
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1answer
43 views

Prove: $\int_{\mathbb{R}^n}e^{-|\mathbf{x}|^n}\,d^n\mathbf{x} = \operatorname{vol}(B(0,1))$ [closed]

Prove: $$\int_{\mathbb{R}^n}e^{-|\mathbf{x}|^n}\,d^n\mathbf{x} = \operatorname{vol}(B(0,1))$$ where $B(0,1)$ is the unit ball in $\mathbb{R}^n$. I'm not sure how to approach this and would ...
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0answers
28 views

How to calculate joint area of circles defined at each point on a continuous curve

Given a curve c (x(s),y(s)) in a 2D space (shown in red) and a radius function r(s) which gives the radius of a circle at that point. The red curve c(s) thus is the union of all centers of the ...
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0answers
18 views

Is it d^3V or dV when integrating for volume (triple integral)?

In a volume integral dxdydz is shortened to dV. I have also seen d^3V. So dcubed V. What is the correct way to write it, or are both correct?
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1answer
16 views

Volume of tanks and speed of water flow

An open rectangular tank of depth $2.4$m has a horizontal base of length $3.8$ m and breadth $2.1$m. A solid metal cylinder of volume $0.865m^3$ rests with its curbed surface on base in the tank . ...