For questions related to volume.

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1answer
29 views

Finding the volume of a solid region

I'm trying to find the volume of the solid region inside the sphere $x^2+y^2+z^2=4$, and the upper nappe of the cone $z^2=3x^2+3y^2$ (I only have to set up the triple integral itself, not evaluate ...
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1answer
35 views

A formula to calculate the partial volume of a capsule or tank?

We are trying to ascertain the correct formula discussed in this post. The volume formula for a capsule (a cylinder with a hemisphere at both ends) is, $$V_c = \pi r^2 H + \frac{4}{3}\pi r^3\tag1$$ ...
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1answer
30 views

Integrate area of function over a tetrahedron

I actually attempted to enlist my professor help on this problem, but what he said was quick and I must not have written everything down because I cannot understand how this problem is supposed to be ...
1
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1answer
23 views

Find the region of integration as defined by two paraboloids

I've been given the following problem, and I'm completely unsure how to go about solving it. $$ \text{Find the volume of the solid enclosed by the}\\ \text{paraboloids } z = 16 \left( x^{2} + y^{2} ...
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1answer
17 views

Find area of exponential function over box-like region

The problem doesn't seem like it should be too difficult, I have a box-like region $B$ defined as: $$ \begin{align} 0 \le x \le 1\\ 0 \le y \le 3\\ 0 \le z \le 2 \end{align} $$ And the function to ...
2
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1answer
68 views

How can I find $\det(A)/\det(B)$, when individual determinants blow up

I am interested in the quantity: $\frac{\det(A)}{\det(B)}$ of positive definite matrices $A$ and $B$. The problem I am running into now is that for large $A$,$B$, (around $200 \times 200$), the ...
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0answers
17 views

Volume of partially filled spherical cap? [on hold]

I have a spherical cap... the plane end (which is of course a circle) is vertically to ground... the radius of the sphere from it we made these cap is R, the distance from center of sphere to the ...
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0answers
20 views

How do you find the volume of a function rotated about the x axis along it's derivative? [closed]

So when you rotate a function, it is usually vertically. How do you rotate it around it's derivative, assuming that volume/area can overlap?
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3answers
99 views

Integrating volume of a sphere with a cylinder “drilled” out of it

Unfortunately, I am stuck again on another integration problem. Famous last words, this should be simple. $$ \text{A cylindrical drill with radius 5 is used to bore a hole through}\\\text{the center ...
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3answers
21 views

Double integrals volume

Find the volume below $z = 5+3y$ above the region $−5 \leqslant x \leqslant 5$, $0 \leqslant y \leqslant 25−x^2$. How do I solve this? I don't know how to make equation to solve this problem. Anyone ...
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1answer
49 views

How to solve this problem using spherical coordinates system?

The question is very simple: Volume inside the solid limited by:$ (X^2+Y^2+Z^2=16), (X^2+Y^2=4)$ using SPHERICAL coordinates system. The final answer however can be checked making a "cylindrical ...
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0answers
13 views

Sorting Bigger Boxes to Smaller Boxes

I am currently working on a Bin Packing program and need to know what would be the most efficient way of getting boxes (arbitrary width, length, and height) to be sorted in the manner below? ...
0
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1answer
12 views

calculate similiar volumes, but with little diffrence

There are to pretty similiar volumes, but there is something in the calculation, that I dont understand clearly and I would be glad if someone help me. The first (Vivani volume): a volume that ...
2
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1answer
45 views

Finding the value of an integral containing $(\ln x)^2$ in the denominator

While reviewing (as an instructor/test editor) a second semester calculus exam, I came across the following problem: Find the volume of the solid created by revolving around the $x$-axis the area ...
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0answers
47 views

How much water could be stored?

We have two water storage tanks--- Tank A and Tank B --- on the roof of the upper storey of our two-storey house. The tanks are cylindrical in shape. Each of the two tanks have a circular opening ...
3
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1answer
105 views

Computing the volume of this weird object,

Let $f: [-1,1] \to \mathbb{R}$ be a continuously differentiable function such that $f(-1) = f(1) = 0$ and $0<f(x)\le 1$ for all $x \in (-1,1)$. Let $S$ be the surface in $\mathbb{R}^3$ obtained by ...
0
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1answer
22 views

Calculate Volume m3?

I'm attempting to calculate the value that is m3 but i don't know how the person got these values. The first row first column is 22mm(width) x 100mm(height) .. then the bold in the second column is ...
1
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2answers
69 views

Complex value for volume, using triple integrals

I'm trying to calculate the volume a hyperboloid, within $$z=0$$ and $$z+\frac 12 x-3=0.$$ The hyperboloid: $$x^2+\left(\frac y2\right)^2-z^2=5.$$ I calculated the projections on $xz$, $yz$, to use ...
0
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4answers
65 views

Calculate the volume bounded by 2 balls.

I need to calculate the volume bounded by: $$x^2+y^2+z^2\:=\:1,\:x^2+\left(y-1\right)^2+z^2=1$$ My solution: Because the volume I want to calculate is symetric, Ill calculate only one half of it and ...
0
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1answer
39 views

Notation: determinant of Jacobian matrix

Given a function $f:\Delta^n \to Y$ from a simplex into a riemmannian manifold. Furthermore given a point $x \in \Delta^n$ we can send an orthonormal basis at $x$ using $D_x f$ to a set of vectors ...
0
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1answer
40 views

Surface area of $y = \sin(\pi x)$, from $x=0$ to $2$, rotated about the $x$-axis.

When I use the surface area formula I get 0, and Wolfram got zero as well when I use the bounds 0 to 2, why is this? However the solution manual uses the integral with bounds 0 to 1.. What is going ...
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0answers
21 views

What is the total volume of wood used for the model?

A person makes a model of a house in construction class. The block of wood for the base measures 6 inches by 4 inches, and is 4 inches tall. He used a triangular prism for the roof, whose ...
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0answers
61 views

What's the largest box I can make out of a sheet of wood?

I'm planning on building a storage box out of sheet wood, and I'd like to know how to cut up my sheet into 6 pieces to form a box with the largest volume possible. Ideally the box should be flush on ...
2
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1answer
16 views

Volume of the symmetric difference between a parallelotope and its translated.

Let $A$ be a n-dimensional parallelotope and $v \in \mathbb{R}^n$ a vector. Is there a formula giving the volume of the symmetric difference $A \Delta (v+A)$?
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2answers
96 views

spherical segment volume

Suppose I have a spherical segment like the one in the picture. I want to find the infinitesimal volume of such a segment. The angle between point A and B is $d\theta$. And the radius of the sphere ...
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1answer
55 views

Is this correct? or incorrect? volume and surface area of cuboids

Is my homework correct? volume and surface area of cuboids http://i.stack.imgur.com/UovRN.png or see below:
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2answers
26 views

Cylinder volume with curved base area

Suppose, we have a cylinder with a flat base area $A$ and height $H$. The volume $V$ of the cylinder is obtained by multiplying the two quantities: $V=AH$. But what happens, when the base surface is ...
0
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1answer
51 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
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0answers
29 views

Calculating the volume bounded by $z = 5$ and $z^2=x^2+y^2$ in 2 ways

I don't understand where is my mistake on calculating the volume by the second way. The volume that I want to calculate is bounded by $z = 5$ and $z^2=x^2+y^2$, so it is the upper part of the cone, ...
1
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1answer
20 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
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1answer
16 views

Evaluate the volume of a solid of revolution using shells.

A cylindrical shell $S$ formed by some revolution about the $y$-axis is given by the equation: $S=2\pi x f(x)dx$, where the circumference $C$ of the shell is $C=2\pi x$, the height of the shell ($H$) ...
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1answer
32 views

Volume bounded by the regions $y= \frac{1}{x}, x=1, x=2, y= 0$ about $x= 3$ using the shell method

Find volume by these bounded regions $y=\dfrac{1}{x}, x=1, x=2, y= 0$ about $x= 3$ (shell method) Not sure what is wrong with my integral here. $$2\pi \int_1^2 \frac{(3-x}{x} \, \mathrm{d}x+ ...
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1answer
51 views

What is the depth of water above the prism?

I have been practising for a math competition and came across the following question: A fishtank with base $100\,\rm cm$by $200\,\rm cm$ and depth $100\,\rm cm$ contains water to a depth of ...
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2answers
38 views

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$.

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$. I tried to graph this using Wolfram Alpha, but it didn't help. I don't know how to start or ...
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0answers
18 views

$Y = e^{-|x|}$,$ x\geq 0$ about $y = 0$. Use disk and shell method

Find the volume bounded by $y = e^{-|x|}$, $x\geq 0$, about $y= 0$. Use disk and shell method. Here is my integral attempt http://www.wolframalpha.com/input/?i=integral+0+to+1+%28y%29%28-lny%29 ...
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0answers
45 views

Volume between $y = 1/x$, $y= 0$, $x=1$, $x=3$ about $y = -1$ (using shell method)

Pretty obvious here that disk method is easy and I got the right answer according to the book with it. However for the last hour I have been trying to use shell method and nothing seems to be working. ...
0
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3answers
72 views

How do I derive the volume of this cup?

How do I derive the volume of this cup? It's been many years since I've taken calulus... So far I've started with the radius of the bottom and integrated that around the circle. Did I start it right? ...
0
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1answer
24 views

Calculating the volume of a solid of revolution about a line.

A figure is formed by revolving the region bounded by $f(x) = \cos{(x)}$ and $g(x) = \sin{(x)}$ from $0$ to $\dfrac{\pi}{4}$ about the line $y=-1$. This figure is formed by integration of two ...
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1answer
98 views

Volume of the intersection of two cylinders

I have two infinite cylinders of unit radius in $\mathbb{R}^3$, whose axes are skew lines. Say that the axis of one is centered on the $x$-axis, and the axis of the other is determined by the two ...
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1answer
17 views

Finding the volume of a cone with and oblique base.

The base of $S$ is an elliptical region with boundary curve $9x^2+4y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. The base of $S$ is ...
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1answer
45 views

Calculating the volume of cutted cone by a plane

I need to calculate the volume bounded by the plane: $x+y+z=5$ and by the cone $z^2 = x^2 + y^2$, som my V that i'm $dv$-ing on it is cutted cone in non simetric way (i can find the equation of the ...
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1answer
54 views

Find the volume of an object using integrals

How am I supposed to find the volume of an object when I know that: $$x^2+y^2\le z^2, \ 0 \leq x, \ 0 \leq y, \ 0 \leq z \leq 1$$
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2answers
49 views

calculating the volume of a room with a lopsided ceiling

Part of my job description requires that I find the volume of a room for calculating air leakage. Normally no problem, but this is an unusual house for many reasons. The main issue I'm having a ...
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0answers
27 views

Shared Volume of Overlapping 3D Cubes/Rectangles

Good afternoon, I have been looking for an approach to figure out the volume where two cubes/rectangles overlap, meaning, I know when they do, I just don't know the coordinates of the volume in which ...
1
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1answer
29 views

Calculate the volume of mass in a burial mound

I am trying to calculate the volume of some burial mounds, but I am embarrassingly poor at math. The shape of an ideal burial mound is most similar to a hemisphere. I have read other people's work ...
1
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1answer
28 views

Minimum Surface Area of a Closed Cylindrical Container

This is a trivial question; but I just want to make sure: A closed cylindrical container has a capacity of $128\pi \,{\rm m}^3$. Determine the minimum surface area. The answer is $96\pi$. Volume of ...
0
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1answer
45 views

Find the volume generated by rotating the given region about the specified line

I am really unsure how to solve this problem. I was hoping someone can give me some helpful hints so I can figure it out. I also looked at some other posts but they weren't helpful Find the volume ...
1
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1answer
23 views

Negative volume of solid of revolution around x axis.

There was a question on my exams yesterday about the volume of revolotion of the solid generated when the circle $x^2+(y-3)^2=1$ rotates around x axis. That moment I thought it would be best if I used ...
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1answer
32 views

Points in a given volume/Area

I have a rectangular prism(3D bounding box) for which i have the point(i.e center of gravity) and the height,width,depth dimensions . Given these parameters, is it possible to find all the points that ...
0
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1answer
13 views

Weight on a specific point

I have a weight of 27tonnes and it has 6 points of contact with the ground. The weight is not evenly distributed, so how do I work out how much weight is on each of the 6 points? I know where the 6 ...