For questions related to volume.

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0
votes
1answer
28 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.
0
votes
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 ...
0
votes
1answer
23 views

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

I have to find the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. So, here we have a cone and a plane "cutting" it. I definitely must do this using some some of coordinate ...
3
votes
1answer
39 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: ...
0
votes
1answer
29 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.
0
votes
2answers
37 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 ...
0
votes
0answers
19 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 ...
2
votes
0answers
108 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 ...
0
votes
1answer
29 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, ...
0
votes
0answers
15 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 ...
0
votes
1answer
24 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
84 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 = ...
-1
votes
2answers
42 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 $A_1$ ...
2
votes
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 ...
0
votes
0answers
38 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 ...
0
votes
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 ...
-2
votes
0answers
15 views

Finding maximal volumes of composite objects

Say you have a can as shown below (sorry about the picture, I couldn't find a better picture of the scenario), If you wanted to maximize the dimensions of the can, or in other words optimize them, ...
0
votes
0answers
19 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 ...
0
votes
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 ...
3
votes
0answers
25 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: ...
-1
votes
1answer
41 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 ...
1
vote
0answers
27 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 ...
0
votes
0answers
17 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?
0
votes
1answer
13 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 . ...
1
vote
1answer
23 views

Find the volume of solid of revolution given two curves?

I am given two curves $y=2-x^2$ and $y=x^2$, from which the bounded region is to be rotated about $x=1$. I have drawn out the shape and see that the two parabolas intersect each other at $(-1,1)$ and ...
1
vote
1answer
38 views

Revolve $y=e^{-x}$ about $x=1$?

I wanted to ask what I think is a fairly simple question for most of you on here! Revolve the curve $$y=e^{-x}$$ bounded by $y=0$, $x=0$, $x=-1$ about the $x=1$ axis. I am supposed to use the method ...
0
votes
1answer
57 views

Finding the derivative of an equation

I am currently doing an investigation in which I am required to design the dimensions of a juice box (can be cube/cuboid) which has the least possible surface area that can hold 200 ml of juice. I ...
6
votes
2answers
124 views

Volume of the Intersection of Ten Cylinders

I'm in Calculus 2, and we were first given the problem to find the intersection of two perpendicular cylinders of equal radius. This breaks down into eight times the volume of a quarter circle ...
2
votes
2answers
31 views

Optimisation of a juice box: finding the least possible surface area that can hold the most volume

I have an investigation which requires me to design the dimensions of a juice box (cuboid) which has the least possible surface area that can hold the most volume. I am not sure as to how I should ...
-1
votes
0answers
33 views

Prove that the figure doesn't have volume

The figure is a set of points $M(x, y, z)$, where $$ 0 \leq z \leq 1 \\ \begin{cases} 0 \leq x \leq 1, 0 \leq y \leq 1 & \mbox{if } z \mbox{ is } Rational\\ -1 \leq x \leq 0, -1 \leq y \leq 0 ...
2
votes
1answer
38 views

Find the volume of the tetrahedron using triple integrals

Find the volume of the tetrahedron with vertices $(0,0,0), (0,0,1), (1,0,1), (0,1,1)$, equation $$x+y=z$$ bounded by $x=0$, $y=0$ and $z=1$. Then $x+y=1$. My integral: is this correct? $$\int_0^1 ...
0
votes
1answer
27 views

Water is being poured into an inverted right conical vessel [closed]

Water is being poured into an inverted right conical vessel whose apex angle at $90^\circ$ at a constant rate of $3 \text{ cm}^3/\text{s}$. At what rate is the water level rising when the depth is $π$ ...
0
votes
2answers
14 views

Volumes by Slicing: why not y-axis?

Find the volume of the solid generated by revolving the region bounded by the graphs of y = x^2 and y = 4x − x about the line y = 6. We should calculate it according to this integral ...
0
votes
0answers
22 views

Alternative formulas for the volume of an n-ball

The volume of an n-ball is \begin{align}\dfrac{\pi^{n/2}r^n}{\Gamma\left(n/2+1\right)}\end{align} However if we define $\alpha = \sqrt{\pi}$, and use the $\Pi$-function, we get ...
0
votes
1answer
32 views

Related Rates. Water is being poured in an inverted right conical vessel

Water is being poured into an inverted right conical vessel whose apex angle at 90° at a constant rate of 3cm^3/s. At what rate is the water level rising when the depth is π cm. Hi,I can easily solve ...
1
vote
2answers
36 views

Volume Under a Plane

Find the volume under z=3x and about he 1st quadrant area in the xy plane bounded by $$x=0, y=0, x=4, x^2+y^2=25$$ $$\int_0^4 \int_0^{25-x^2}3x dxdy = 98$$ Not sure if this is correct. I was ...
0
votes
0answers
22 views

The Area of Something Known Only in Volume

In my worldbuilding, I have created a plain of lava that extended to a volume of 59 to 77 million cubic kilometers and 4500 meters at the thickest. As I understand it, the basic formula of volume is ...
0
votes
1answer
31 views

Calculating volume of solid

How would I go about calculating the volume, $V(M_1)$, of $M_1$? $$M_1 = \{(x,y,z) \in \Bbb R^3 : (x,y) \in D,\ 0 \leq z \leq yx2 + y^3\},$$ $$D =\{(x, y) \in\Bbb R^2 : |x| \leq y, 1 \leq x^2 + y^2 ...
-2
votes
0answers
21 views

Having trouble calculating volume of solid

Having trouble calculating volume of a solid bounded as: M1 = (x,y,z) ∈ R^3 : (x,y) ∈ D, 0 ≤ z ≤ yx2 + y^3. D =(x, y) ∈ R^2: |x| ≤ y, 1 ≤ x^2 + y^2 ≤ 4 Im doing as follows so far: ...
6
votes
3answers
191 views

Diagonals of squares on curved functions

I just came across an integration problem. It is very easy to plug numbers into the steps of the solved problem and arrive at the right answer, but I don't understand one of the choices of formulas ...
0
votes
1answer
17 views

Jacobian matrix for ellipsoid

ive been asked to fine the jacobian matrix for an ellipsoid $$x^2/a^2 + y^2/b^2 + z^2 / c^2 = 1$$ ive been looking online for the parametric equations and i get two different answers ...
1
vote
2answers
50 views

Inner volume of quadric surface

Can anyone help me work out the inner volume of the following quadric surface $x^{2/3} + y^{2/3} + z^{2/3} = 1$? Edit: Adjusted the title and the line above but have left the rest of the question ...
2
votes
1answer
46 views

maximum volume of a box inside an ellipsoid

What is the maximum volume of a box that can be placed inside an ellipsoid $\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$ The volume of a box is $V=xyz$ so I need to find $x,y,z$ with respect to ...
0
votes
2answers
61 views

Help in calculating this tedious volume. [closed]

The base diametre of glass is 20% smaller than the diametre of rim. The glass is filled to half the height.What is the ratio of empity to filled volume?
1
vote
0answers
24 views

How to calculate the volume of revolution around $x=3$ for the region bounded by x, y axis and $\sqrt{x}+\sqrt{y}=1$?

So I intend to do it in both shell and disk ways. Let's first use shell method (formula: $\int 2\pi x(f(x)-g(x))dx$): Since $x=3$ is the major axis, and $y=(1-\sqrt{x})^2$, we have $V=\int_0^1 2\pi ...
0
votes
0answers
60 views

What is the cap body produced by the unit sphere?

For ecach number a > 1, let C(a) be the cap body produced by the unit sphere in E^(3) and the points (+-a,0,0). Calculate the volume, surface area and mean width of C(a). For this question, I don't ...
2
votes
1answer
43 views

Intuition for volume of a simplex being 1/n!

Consider the simplex determined by the origin, and $n$ unit basis vectors. The volume of this simplex is $\frac{1}{n!}$, but I am intuitively struggling to see why. I have seen proofs for this and am ...
0
votes
0answers
37 views

Calculate the volume bounded by the surface $(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}$

I need to solve: Calculate the volume of the solid bounded by the surface $(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}$ and not sure on how to ...
1
vote
0answers
21 views

Limit of uniformly converging volume-preserving homeomorphisms

Definition A map $f\colon \mathbb{R}^n \to \mathbb{R}^n$ is volume-preserving if for every (Lebesgue) measurable set $V\subset \mathbb{R}^n$, $\mathcal{L}^n(V) = \mathcal{L}^n(f(V))$. I am wondering ...
2
votes
1answer
43 views

Find the volume of the areas bounded by following:

$(x^2+y^2+z^2)^2=a^2(x^2+y^2-z^2)$ with $a = const$ $z=x^2+y^2, z^2=2(x^2+y^2), xy=a^2 , xy=2a^2,x=2y, 2x=y$ and $x > 0, y> 0$ First of all, I have never heard of this first geometric shape ...