For questions related to volume.

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volumes and integrals help [on hold]

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$ and $y=x^4$ and $y=16$ and $x=0$ about the $y$-axis.
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1answer
25 views

Find the volume of the region bounded by $z=0,x+z=4$ and $x^2+y^2+z^2=4$. [on hold]

Find the volume of the region bounded by $z=0,x+z=4$ and $x^2+y^2+z^2=4$. My problem:I am unable to solve it I can not imagine how it looks and what are the limits of integration. Thanks
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1answer
24 views

Derivation of the volume of a prism.

How can we prove that the volume of a prism with any plane region as its base is the area of its base multiplied by its height? By volume I mean the measure of the quantity of three-dimensional space ...
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2answers
35 views

Finding the volume of revolution using the method of shells

I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about $x$-axis using the shell method. I applied the method and I got $15864/5$ ...
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0answers
18 views

Evaluate the volume of the solid determined by $x,y,z\in[0,1],z\geq xy$

Evaluate the volume of the solid determined by $x,y,z\in[0,1],z\geq xy$ My calculations: $$V=\int_0^1 \int_0^1 \int_{xy}^1 dz\,dy\,dx=3/4$$ Is this correct?
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1answer
24 views

Center of mass of a barrel partially filled with grain

Bubba has a barrel in the shape of a cylinder of mass 39.4 kg. The barrel has a diameter of 62.4 cm and is 1.32 m tall. He fills the barrel to a depth of 49.5 cm with loose packed grain that has an ...
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1answer
16 views

Volume of a region after some transformation.

Consider $$R=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq 2\}$$ and the transform $$T:(x,y,z)\to(x,y+\tan \alpha z,z)$$ where $0<\alpha<\pi$ Then what is the volume of $T(R)$? I tried myself but I ...
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1answer
21 views

Calculate the volume of intersection of $x^2+y^2+z^2=4$ and $r=2\cos \theta$ by using cylindrical coordinates.

Calculate the volume of intersection of $x^2+y^2+z^2=4$ and $r=2\cos \theta$ by using cylindrical coordinates. My try:Intersection will be a cylinder $x^2+y^2+z^2=4\implies r^2+z^2=4$ Then ...
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0answers
15 views

Calculating volumes using integral.

Given $y=x,y=0,x=2$ and $x=7$. Calculate the volume6 about $x=1$. I just need to get the concept right. Please tell me what mistake I did here. The region looks like a trapezium right? From $y=0$ ...
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0answers
20 views

Excercise: Find the volume of the parallelepiped

Find the volume $V$ of the parallelepiped whose four adjacent vertices are the points: $A = (−2, 1, 0)$, $B = (2, 3, 2)$, $C = (1, 4, −1)$, and $D = (3, 6, 1)$. I know how to find it with three ...
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1answer
17 views

Combining two liquids with different weights to achieve a desired volume and weight

I have two liquids - water and alcohol, each liquid has a different mass Water weighs 1 gram per ML Alcohol weighs 0.5 gram per ML (just for the sake of the example) I wish to combine these ...
0
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1answer
54 views

Finding the volume and moment of inertia

Considering an elliptic torus. The boundary equation is: $$\frac{(\sqrt{x^2+y^2}-c)^2}{a^2}+\frac{z^2}{b^2}=1$$ about the $z$-axis. Find the volume of this object Find the moment of inertia about ...
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2answers
34 views

How many milliliters to fill cone

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more milliliters of liquid need to be added in order to fill the ...
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0answers
23 views

Using polar co-ordinaries on an integral whose domain is a disk not centred at the origin

I need to find the volume of the region in $z > 0$ that lies within the cylinder $x^2 + y^2 = 2x$ and is bounded by the cone $z^2 = x^2 + y^2$. I have been struggling to set up the integral for ...
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2answers
35 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
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1answer
23 views

Using differentials with volume of a cube

my question is The volume of a cube is increased from 1000 cubic centimeters to 1156 cubic centimeters. Use differentials to determine. the side length of the cube increases by? the surface area ...
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0answers
21 views

Help required in kernel density estimation

In http://www.csc.kth.se/utbildning/kth/kurser/DD2427/bik08/LectureNotes/Lecture6.pdf Slide#3, the problem stated is that in $k$ nearest neighbor method assuming the d- dimensional data points are ...
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1answer
34 views

Maximizing volume in Calculus

An open box is to be created from a flat piece of cardboard 36 inches square by cutting a square from each corner and then folding up the edges. How long should the side of the square being cut out in ...
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0answers
12 views

Calculate the centroid and mass moment of intertia of a prism

I am wondering if there is a "trick" I can use to ease the equation level for this problem. Say I have a solid (constant density) prism which is bounded by the following 6 "planes" (I use subscript ...
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1answer
25 views

Calculating the volume of a solid of revolution about the y-axis question

I am trying to find the correct answer for the volume of solid of revolution generated by revolving the region bounded by the following curves around the y-axis: $$ y = \sqrt{25-x^{2}}; x = 3; y = ...
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2answers
36 views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
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1answer
209 views

Why is it enough to prove the sentence?

I am looking at the proof of the theorem that for any rectangle the outer measure is equal to the volume. At the beginning of the proof there is the following sentence: It is enough to look at the ...
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3answers
67 views

Integral using height to find volume

How do you find the volume of a "pit" which is circular in horizontal cross-section, and parabolic in vertical cross-section using height by "sticking". "Sticking" is when we insert a dipstick through ...
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1answer
28 views

Find the volume rotating the region enclosed by the curves $xy=1$, $x = y^{1/2}$, and $y = 2$ about the $y$-axis

Find the volume rotating the region enclosed by the curves $xy=1$, $x = y^{1/2}$, and $y = 2$ about the $y$-axis I've looked up solutions but nothing looks like my problem. I did draw a ...
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1answer
25 views

Adjusting weight of a body of water by substituting part of it with a lighter liquid

As a heavily simplified example of my problem: Water weighs 1 gram per ML Alcohol weighs 0.5 gram per ML (not true of course, but humour me) I have 100mls of water, so this has a weight of ...
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0answers
26 views

Volume of the set of solutions of two linear inequalities on the simplex

I need an analytical formula for the volume of the set of solutions of two linear inequalities on the N-dimensional simplex $\Omega$. Two be more precise, if $u$ and $v$ are two vectors of ...
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1answer
18 views

Find the volume of the solid obtained by rotating the region about the y-axis

Find the volume of the solid obtained by rotating the region bounded by $y=5\sin(5x^2)$, $0 \le x \le \sqrt{(\frac\pi5)}$ about the $y$-axis. I get the wrong answer using the cylindrical shell ...
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2answers
31 views

Find the volume of revolution

Calculate the volume of the solid generated by rotating the region $\{x=1,y=2,y=x\}$ along the y-axis. My problem:My region is a triangle,how I can calculate this?I know the formula for curvilinear ...
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0answers
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Given the equation for a surface, how to find enclosed volume?

Suppose we give an equation of the form $f(x_1,x_2,..., x_n)=C$, with $f$ a smooth function, and assume this is such that defines a closed surface in $\mathbb{R}^{n+1}$. Assume also that the equation ...
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1answer
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Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
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1answer
25 views

Calculating volume by area and thickness

I have an irregular hexagon that is 1 mm thick. The total of the area of the hexagon is 114.335 sqaure cm and as I said the thickness is 1 mm. How do I calculate the volume?
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1answer
22 views

find out the volume of solid removed?

i have sphere that has an equation $$x^2+y^2+z^2=1$$ a cylindrical hole $x^2+(y-1/2)^2$=$1/4$ is cut through it . find the volume of the portion cut. i don't know what to do, i was thinking of using ...
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1answer
58 views

Word problem for polynomial equations: Volume of a cylinder given relation between radius and height

A cylinder has a volume of 324 cm^3. If the radius of the cylinder is 1 cm more than twice the height, find the dimensions of the cylinder. I know that formula for the volume of cylinder is $V=\pi ...
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1answer
28 views

What is the volume of an arbitrary intersection of a cube and a cylinder?

I need to find the volume of an arbitrary intersection of a cube and a cylinder. the sides of the cube ($a$) will always be less than the diameter of the cylinder, such that a cube can fit fully ...
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2answers
58 views

Find the volume formed by rotating the enclosed region

Find the volume formed by rotating the enclosed region $y=4\sqrt{x}$ and $y=x$ about $x=17$ I have tried plugging everything into the formula but I can't seem to get the right answer. How do I solve ...
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1answer
24 views

Compute volume of tetrahedron using a triple integral

I'm trying to compute the volume of a tetrahedron with the vertices (0, 0, 0), (0, 0, 1), (2, 0, 0), (0, 2, 0). It needs to be done using a triple integral. Not allowed to use "det" or other ...
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1answer
14 views

shell method rolving twice

let A is the region enclosed by $x=1, y=1, y=1-x$. then suppose S is the solid obtained by revolving A about y-axis. Find volume of soild obtained by revolving S about the x-axis. I've used shell ...
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0answers
28 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between ...
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0answers
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Volume of intersection between two equal cones with parallel axes

The two infinite cones (nappes) (each 45-degree wide) have parallel axes. They are oriented in opposite directions, and the top of one is inside the other, so that the common volume V is finite. How ...
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0answers
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Check if surface area is big enough to create volume

Given volume and surface area only, how can I check whether the surface area is enough to create the specified volume? If the lowest area to volume ratio suggests that a sphere is the optimum ...
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1answer
28 views

How to set up an integral by these conditions?

I've got these surfaces: $$ z = 0\\ z = 4 - y^2 $$ And a cylinder: $$ x^2+y^2=4 $$ I need to find the volume enclosed by these figures. As far as I understand the limits of integration for $z$ are ...
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1answer
20 views

How to reduce 3 dimensional optimization to 2 dimensions?

I am trying to minimize the surface area of a parallelepiped of unit volume. Using Volume = xyz(1 + 2cos(a)cos(b)cos(c) - cos^2(a) - cos^2(b) - cos^2(c))^1/2 = 1 where x,y,z are edge lengths and ...
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0answers
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How to find the parallelepiped of unit volume with minimal surface area?

Is it best to approach this problem using edge lengths and the angles between them? I am trying to reduce the problem to two dimensions, although I haven't successfully done so yet So I have Volume ...
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0answers
39 views

Volume of a rotation solid

A student has to calculate the internal volume of a dome whose curvature is given by the equation $2 x ^ 2 + y ^ 2 = 32$, where $y\geq 0$. The maximum diameter of the dome at the base is $8$ meters. ...
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2answers
45 views

Proving the Volume of an Ellipsoid

This is the question: The solid generated by rotating the region inside the ellipse with equation $$ \left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 = 1 $$ around the $x$-axis is ...
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1answer
26 views

Integrals in Reverse

I'm asked what solid this integral represents(the integral is used to obtain the volume of a solid, we are given this). I see that since we have the 2pi, this is probably a volume obtained by using ...
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1answer
22 views

Finding volume of solid using two methods

The question is: The region bounded by $y=\frac{1}{x}, y=0, x=1, x=2$ is rotated about the $y$-axis, thus creating a solid. Compute the volume using the Shell and Slicing method. This is what I have ...
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1answer
10 views

Find the dimensions of a square piece of cardboard given data of it folded into a square (cubic inches, etc.)

A box with a square base and no top is to be made from a square piece of cardboard by cutting 6 in. squares out of each corner and folding up the sides. The box needs to hold 1000 in3 . How big a ...
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1answer
33 views

Setting up integral for volume of the solid

How would I set up an integral for the volume of the solid bounded between these two curves: $$y=x$$ $$y=\frac{2x}{1+x^3}$$ Rotated about x=-1. And these two curves: $$y^2 - x^2 = 1$$ $$y=2$$ ...
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1answer
20 views

Using the shell method, find the volume of the solid by rotating the region bounded by the given curves

$$x=y^2+1$$ $$x=2$$ about y=-2 How would I set this up? This is what I have so far: $$V = \int_0^2 2 \pi (y+2)(y^2+1) dy$$ I am almost certain this is wrong. Especially with the limits of ...