For questions related to volume.

learn more… | top users | synonyms

1
vote
1answer
42 views

Volume of the solid bounded by the planes (Checking the limits of the integral)

Find the volume $V$ of the solid bounded by the planes $x+y-z=3$ and $z=0$, and the cylinder $x^2+\frac{y^2}{4}=1$. My calculations give Polar $$V = \int_{\theta=0}^{\theta=\pi/2} \int_{r=0}^{r=1} ...
0
votes
1answer
32 views

Volume of the pyramid…

I have such a problem from geometry: Five edges of a regular triangular pyramid have the length of $6$ $dm$, but the sixth- $4$ $dm$. Determine the volume of the pyramid. For me the problem is quiet ...
1
vote
3answers
39 views

For solid volumes, why does the Integral behave as a summation?

When you take a definite integral, you can think about calculating the area under the curve (via Riemann rectangle slices approximation) Now, when you take the volume of a 3D object, you sum the ...
0
votes
0answers
25 views

Volume by double integration?

Suppose that $h<a<0$. Show that the volume of the solid bounded by the cylinder $x^2+y^2=a^2$, the plane $z=0$ and the plane $z=x+h$ is $V=\pi a^2h$. I'm having a very hard time with ...
2
votes
3answers
46 views

Maximum volume of a box given perimeter and surface area

What would be maximum volume of a rectangular box with a given perimeter $P$ and surface area $S$? I tried to solve following equations, where $l$ is length, $b$ is base, $h$ is height, $P$ is the ...
2
votes
2answers
46 views

Volume of a parallelpiped from its sides and diagonals?

If a parallelogram has sides of length $a$ and $b$, and diagonals of length $d$ and $e$, then we can find its area in the following way. By the polarization identity, we have $a b \cos\theta = ...
0
votes
0answers
14 views

Cross sections of a cone

If the ratio of the areas of the triangle and rectangle below is 1:2 Then why isn't the ratio of volumes of a cone and the smallest cylinder that contains it 1:2? If each "slice" has a 1:2 ratio ...
0
votes
1answer
22 views

A volume integral question

When the region enclosed by the graphs of $y = 2x$ and $y = 6x-x^2$ is revolved around the $y-axis$, the volume of the solid generated by is given by?
1
vote
1answer
21 views

Definite integral in spherical coordinates

I want to compute the volume enclosed by the sphere $x^2+y^2+z^2\le4$ with $0\le z\le 1.$ If I use the rings method I get: $$\pi \int_0^1 \left( \sqrt{4-z^2}\right)^2dz=\frac{11}{3}\pi $$ If I set ...
0
votes
1answer
26 views

Finding volume of convex polyhedron

I am trying to compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0),(0,2,0),(0,0,3) and (10,10,10). I am supposed to use a triple integral but am struggling with how to set it ...
1
vote
1answer
18 views

How do I graduate a cylinder glass in milliliters?

Today I decided to cook something, but then I realized there is a critical item missing − a measuring glass. Being a programmer and all, I decided this wouldn't be much of a problem, as I could ...
0
votes
2answers
44 views

How to find the volume of the solid

I need some help on this question, I even know nothing about the shape. How can I decide the range of x,y,z? Thanks a lot!
0
votes
0answers
10 views

How to show that the volume contained by a surface is equal to a double integral with a vector field.

I am given a curve $C$ which is parametrized by: $x(t)=Cos(t)$ and $y(t)=Sin(2t)$ for $-\frac{1}{2}\pi\leq t \leq \frac{1}{2}\pi$ Let $S$ be the surface you get if you revolve $C$ around the ...
0
votes
0answers
23 views

find the volume of the region using triple integrals using cylindrical coordinates

The volume of the pyramid defined by $(0,0,0)$, $(2,0,0)$, $(0,1,0)$ and $(0,0,4)$. Calculate: $\displaystyle\iiint(2+z^2)\,dV$ The limit of the radius is where I am stumped, since the $xy$ ...
0
votes
1answer
13 views

Volume of Minkowski sum of a point and a hypercube

Let $A$ be a single point and $B$ a unit cube in $\mathbb{R}^n$, what is then the volume $\lambda \mapsto \mathrm{Vol}\big((1-\lambda)A + \lambda B\big)$? I am not exactly sure, what the set ...
0
votes
0answers
53 views

Elementary Calculus

Trying to help a friend with a basic calculus I problem. Find the volume of the figure enclosed by planes perpendicular to the x axis, $x=-4$, $x=4$. The cross sections are squares with base ...
0
votes
1answer
37 views

Volume of unit n-dimensional ball, definite integal

As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true: ...
1
vote
1answer
33 views

Volume of the Region bounded by $y = 2x^2 +2z^2$ and the plane $y=8$

I have the find the volume of the region bounded by the paraboloid $y = 2x^2 +2z^2$ and the plane $y=8$. Is the volume (using triple integrals) just ...
1
vote
0answers
15 views

Find the volume between a hyperboloid and a cylinder

I'm trying to find the volume bounded by the graphs of $z = 0$ and $z = h$, outside of the cylinder $x^2 + y^2 = 1$, and inside the hyperboloid $x^2+y^2-z^2 = 1.$ I have tried to use cylindrical ...
0
votes
3answers
61 views

In sphere $r \propto \frac{1}{A}$! How is this possible? What's the wrong here?

Surface area $A$ and volume $V$ of a sphere of radius $r$ are \begin{eqnarray} A=4\pi r^2,\\ V=\frac{4}{3} \pi r^3. \end{eqnarray} But then \begin{align} \frac{V}{A} & = \frac{r}{3}\\ ...
0
votes
1answer
32 views

Find volume of the cap of a sphere of radius R with thickness h

I have to determine the volume aka the formula for the volume for this spherical cap of height h and the radius of the sphere is R. ...
0
votes
1answer
34 views

Volume of a 4-sphere of radius 1

I want to express the volume of a hypersphere of radius 1 with a quadruple integral. I know that the equation of the 4-sphere is $x^2 + y^2 + w^2 \le 1$. However, I don't know how to proceed and I'm ...
1
vote
1answer
35 views

Volume of the solid bounded by $4x^2 + y^2 =4$ , $z=o$ and $z=x+5$

I need to find the volume of the solid bounded by the cylinder $4x^2 + y^2 =4$ , $z=o$ and the plane $z=x+5$. I know that $0 \le z \le x+5$ , $-2 \sqrt{1-x^2} \le y\le 2 \sqrt{1-x^2}$ and $-1 \le ...
0
votes
1answer
29 views

Volume of the solid bounded by $z=x^2 + y^2$ and the sphere of ratio R centered at the origin

I'm trying to find the volume of the solid bounded by $z=x^2 + y^2$ and the sphere of ratio R centered at the origin. My problem is that I can't seem to find the proper limits. My thoughts: the solid ...
0
votes
1answer
83 views

Surface area within a cube

What is the surface are of the largest sphere that fits inside the largest cylinder that fits inside a cube with surface area 8?
0
votes
1answer
18 views

Triple integral - wedge shaped solid

Find the volume of of the wedge shaped solid that lies above the xy plane, below the $z=x$ plane and within the cylinder $x^2+y^2 = 4$. I'm having serious trouble picturing this. I think the z ...
0
votes
0answers
16 views

Figuring out volume of partially filled frustrum without area of one side

I have a frustum of a pyramid, which is partially filled with any sort of material (for reference, the frustum is the bottom of a bulk storage silo). So the bottom dimensions a and b are known (the ...
0
votes
1answer
43 views

Volume of a curve rotated around the y-axis

"Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves $y = 3+2x-x^{2}$ and $y+x=3$ about the y-axis. Below is a graph of the bounded ...
1
vote
1answer
34 views

What happens when one side of a cube is multiplied by 7?

I have a question that seens obvious, but I know the answer can't be that simple. What happens to the volume and surface area when all sides of a cube is multiplied by 7? At first I assumed that ...
0
votes
2answers
71 views

How many Scythians were there?

I was doing a maths test yesterday and the last question on the exam was as follows: $2500$ years ago, a Scythian king called Ariantas ordered every one of his subjects to bring him an arrow head. ...
1
vote
1answer
32 views

Using integration and polar coordinates to find the volume of a torus

How would I find the volume of the body formed by revolving the circle $r = f(\theta) = \cos\theta$ about the line $\theta = \frac{\pi}{2}$ ? (This is the circle of radius $1$ centered at $(0,1)$ ...
1
vote
2answers
41 views

Volume of a “tent”

In $(x,y,z)$-space, the ground is the $(x,y)$-plane $z=0$. Above the ground is constructed a giant tent whose height over $(x,y)$ is $$ h(x,y)=z=\frac{100}{1+(x^2+4y^2)^2} $$ Find the volume enclosed ...
1
vote
2answers
34 views

Volume of a solid in $xyz$-space

Calculate the volume of the solid in $xyz$-space bounded by the surfaces $$ \displaystyle z=\frac{1}{x^2+y^2+1} and\:z=\frac{1}{x^2+y^2+4} $$ I haven't done triple integrals for a long time. But it ...
1
vote
1answer
71 views

How deep would the water be?

The lake created by Hoover Dam on the Arizona-Nevada border has a capacity of 31,250,000 acre-feet. If the whole lake suddenly flooded the Mojave Desert (area 15,000 sq. miles) how deep would the ...
0
votes
1answer
30 views

Highest Volume/Area Ratio

Given a fixed volume of a solid, what would be the shape of such solid that would minimize the its surface area? How to determine it? I thought about it, but I cannot find an algorithm that doesn't ...
0
votes
2answers
114 views

volume inside sphere but outside hyperboloid

I am trying to find the volume inside the sphere $x^2 + y^2 + z^2 = 9$, but outside the hyperboloid $x^2 + y^2 - z^2 = 1$. by using a triple integral. for some reason i just cant seem to come up the ...
1
vote
2answers
35 views

Volume of a simplex

Find the hypervolume of the hypersolid in 4-space $\mathbb{R}^4$ consisting of the points $(w,x,y,z)$ that satisfy $w\ge0,x\ge0,y\ge0,z\ge0$ and $w+2x+3y+4z\le6$ This is a problem from our ...
1
vote
1answer
36 views

Gabriel's horn about the y-axis

I'm trying to figure out if there is a solution to finding the volume of the shape that is formed by rotating $f(x) = \frac 1x$ about the y-axis. Not the x-axis as in Gabriel's horn, but the same ...
2
votes
1answer
62 views

Find the volume of the solid bounded by the surface given in spherical coordinates by $R = 4-3\cos(\phi)$.

It is worth noting that $R$ in this case denotes the distance from origin to a point $P$ in space. You may be more familiar with $\rho$ instead of $R$. Here is my attempted solution: I am assuming ...
2
votes
1answer
36 views

Volume of ellipsoid using Linear Algebra

Can someone tell me how to find the volume of an ellipsoid of dimension $\mathbb{R}^3$ by using linear algebra? I know the formula is $\frac{4}{3}\pi abc$. I am given the equation ...
0
votes
1answer
26 views

Volume using shells

I'm working on a problem of finding volume between two functions using the shell method. The functions given are $f(x) = 2x - x²$ and $g(x) = x$. It is reflected across the x-axis. I solved this ...
3
votes
1answer
51 views

Set up the limits and evaluate $\int_{}^{}\int_{}^{}\int_{}^{}ydv$

Set up the limits and evaluate the integral. $$\displaystyle\int_{}^{}\int_{}^{}\int_{}^{}ydv$$ $$G$$ G is the region enclosed by the plane $z = y$, $xy-plane$ and the surface $y = 1 - x^2$ I need ...
1
vote
1answer
26 views

Volume of Solid with Revolution about y-axis

Given two curves, find volume of the solid that results when the region enclosed by the curves is revolved about the given axis. Curves are: $-y^2 + 2 $ $x = y$ Axis is $x = -2$. Having found the ...
1
vote
1answer
48 views

How can you measure out six liters of water?

You want to prepare a tub for your favorite game, dunking for apples. You have two buckets. One of the buckets will hold $4$ liters of water and the other will hold $9$ liters. There are no ...
0
votes
0answers
18 views

Cylinder transform into Cube with the same volume

I want to convert a cylinder shape to a cube shape with the same volume but with 1 height. If a cylinder with a diameter of 2 and length of 2, what are the length and width of a cube when the given ...
1
vote
1answer
113 views

volume of a solid when the graph of $f(x)= \sec^2x$

Find the volume of the solid of revolution obtained when the graph of $f(x)= \sec^2x$, from $x= -\frac{\pi}{4}$ to $x= \frac{\pi}{3}$, is rotated about the x-axis. Give your answer to four decimal ...
0
votes
1answer
26 views

Fractional change in volume from scale-factor

I was given the following question which I am unable to get a seemingly correct answer from: A body expands linearly by a factor $\alpha$ due to an increase in temperature. Because of the ...
0
votes
1answer
52 views

Revolve the region arund the y-axis and find the volume

The region between $y = sin(x^2)$ and the x-axis for $0 \le x \le \sqrt{\pi}$ is revolved around the y-axis. Find the volume of the resulting solid. I can get all the way to the integral: $ ...
0
votes
0answers
32 views

Derivation of expression for Concentration in a small sphere located inside a bigger sphere

I have this problem that is related to diffusion of molecules inside a sphere illustrated in Fig. 1. It goes like this: Given a scenario where there is a concentration of some molecules inside the ...
0
votes
0answers
23 views

Calculate the fraction of volume of a rectilinear grid cell within some radius of the origin

I have a sphere (radius $R$) on a rectilinear grid. Some cells intersect the edge of that sphere, call them 'edge cells'. Designate a given cell by indices $[i,j,k]$ which refer to the lowest-index ...