For questions related to volume.

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1answer
57 views

Why is it incorrect to integrate by $d(2x)$?

I tried to prove the volume of a cone. If you let the radius be $r$ and let the height be equal to the radius, then all you need to do is integrate the area of a circle with radius $r$ by $dr$. ...
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0answers
25 views

Calculate the area of a solid of revolution

So the subject title is self-descriptive. My question is how can I calculate the area of a solid of revolution with the information below: ...
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1answer
34 views

Solid of revolution how to set the regions

I am stuck in this exercise, I cannot get the right answer. The exercise is the following: Rotate around $y = 1$ the region that is between $y=1$, $x=3$, $y=x^\frac{3}{2}$ and the x-axis. As far as ...
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0answers
5 views

Get LWH with volume ≥ X and smallest possible surface area

The formula for volume of a rectangular prism is $l\cdot w\cdot h$, and surface area is $2(wl + hl + wh)$. If I already have the volume (ie 20m²) as $X$, what are the optimal values for $l$, $w$, and ...
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0answers
9 views

cone that holds 2 ounces

if cutting a central cross section of a cone yields an angle of 22 degrees at its point, what does the height have to be to hold 2 ounces or 3.6 cubic inches of water? I believe the answer should be ...
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0answers
13 views

Why it is the same when a curve is rotated 4 right angles or 2 right angles about the y-axis? [on hold]

i realise when the question ask for volume of revolution form when rotated 2 right angles or 4 right angles about the y-axis, both yield the same result. however, this is not so when rotated about ...
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1answer
28 views

Choosing a technique for solids of revolution

Is there a heuristic to choose between the disk method and the washer method? To take the simplest example $y=x$ can be revolved around the $y$ axis using $R=x$ or $r=x$ and $R=$ a constant $C$.
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0answers
22 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : ...
2
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2answers
39 views

Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
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1answer
35 views

Finding Moment of Inertia of a Rugby Ball

I am asked to compute the moment of inertia about the $z$-axis of a rugby ball in terms of its total mass. A rugby ball surface is given by the ellipsoid: $$\frac{x^2}{4} + \frac{y^2}{4} + ...
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1answer
14 views

Max volume of a cylinder

Find maximum volume of a cylinder of which the sum of height and the circumference of the base does not exceed 108 cm. How to solve this? Precisely what is the expression that should be minimized? How ...
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0answers
16 views

Volume and surface of knock out drum

I have to calculate the volume and the surface of some KO drums (knock out drum). To avoid ambiguous understandings here's a picture of one: http://www.zamilsteel.com/ped/images/projects/11.jpg I ...
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4answers
436 views

Finding Volume of Rugby Ball

I am asked to find the volume of rugby ball whose surface is given by the ellipsoid: $$\frac{x^2}{4} + \frac{y^2}{4} + \frac{z^2}{9} = 1$$ I am having trouble figuring out which coordinate system I ...
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0answers
23 views

Finding the volume of a sculpture

A sculpture is given by the region bounded by the two surfaces in $(x, y, z)$ space defined by $z=(x^2 +y^2-1)^2$ and $z=4$. What is the volume of the sculpture? I attempted to find the volume by ...
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0answers
33 views

Volume of 3D shape with parallelogram as base

I want to find the volume of a 3D shape with parallelogram as base and it has like a half ellipsoid on it. The following figure has a tube with bulges on it. I want to find the volume of each bulge ...
1
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1answer
41 views

Finding the volume of the set of all $x \in \mathbb R^4$ satisfying $x^t A x \leq 1$ for a symmetric matrix $A$

Find the volume in $\mathbb R^4$ of the set of $x$ with $x^tAx \le 1$. You may use the fact that the volume in $\mathbb R^4$ of the set of $x$ with $|x|^2 = x^tx\le 1$ is $\frac{\pi^2}{2}$. My ...
0
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1answer
29 views

Volume of solid generated by rotating about an axis

What is the volume of the solid made by rotating the region bounded by $y=x^2+1$ and $y=-2x+3$ about the $x$ - axis. Please show the process using: (i) disks/washers (ii) cylindrical shells
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1answer
15 views

Co-ordinate analysis

Coordinates $ (\alpha, \beta, R) $ with $ -1 \ge \alpha,\space \beta \le1,\space R \lt 1 $ are related to Cartesian co-ordinates $ (x, y, z) $ via $ x= R \alpha, \space y= r \beta $ and $ \space ...
2
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3answers
54 views

Calculate the volume bounded by the surfaces

Calculate the volume of the solid bounded by the surfaces $$\begin{aligned}z&=4x^2+4y^2, \\ z&=x^2+y^2, \\z&=4.\end{aligned}$$ I made an equation of $4x^2+4y^2=4-x^2+y^2$ and solved it to ...
-2
votes
2answers
32 views

Express the volume as a function of the height [closed]

A conical cup is 9 inches high and its radius across the top is 2 inches. If the cup contains liquid, express the volume of the liquid as a function of the height of the liquid.
2
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2answers
54 views

Finding the volume of a rectangular-based pyramid using calculus?

I have a maths test coming up, and I just can't seem to solve a question on finding volumes of solids (via integration). Here's the question: Find the volume of a pyramid with height h and ...
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0answers
30 views

Finding Volume Integration Method

One of the applications of integration is to find volume of solids. But How do i understand to use which method to be used ? For example Cross section, washer, shell etc
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1answer
42 views

How many total square feet is my driveway? [closed]

I need to know how many total square feet my concrete driveway is. I do not have an exact width or length, but the driveway is 23.5 square yards at 4 inches deep. How many total square feet of ...
1
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1answer
37 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
46 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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2answers
56 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 ...
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1answer
24 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
20 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
votes
1answer
69 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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3answers
107 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
24 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 ...
1
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1answer
57 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
15 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
49 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
50 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
votes
1answer
110 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
28 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
71 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 ...
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4answers
67 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
43 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
49 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
26 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
62 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
votes
1answer
18 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
101 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 ...
0
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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, ...