For questions related to volume.

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How fast does the water level decrease in a cylindrical tank?

Is this solution correct? What I know is that the volume of the tank is $V = \pi r^2 h$, where r and h are in meter. Water is drained by a rate of $2,7\frac{m^3}{min}$. How fast does the water level ...
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18 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 ...
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1answer
53 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
33 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
22 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
28 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
21 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
11 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
24 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
33 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
147 views
+100

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
63 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
25 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
23 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
25 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
17 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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21 views

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
14 views

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
24 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
48 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
16 views

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
27 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
29 views

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
43 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
21 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 ...
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1answer
40 views

Volume generated by lemniscate revolving about a tangent at the pole.

The lemniscates $r^2 = a^2\cos2\theta$ revolves about a tangent at the pole. What is the volume generated by it ? Please explain in detail. I found a couple of answers on finding surface areas, ...
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1answer
33 views

Use the shell method to find the volume of the solid obtained by rotating the region bounded by the given curves

$$y=\sqrt{x}$$ $$y=0$$ $$x=1$$ about x=-1 Does this set up look alright: $$V = \int_0^1 2 \pi (x+1)(\sqrt{x}) dx$$
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1answer
34 views

Finding volume by rotating $\cos x$ and $\sin x$ about $y=-1$, $x\in \bigl[0,\frac{\pi}{4}\bigr)$

I'm working on a problem that is asking for the volume of $y=\cos x$ when rotated about the line $y=1$, with a restricted domain of $\bigl[0,\frac{\pi}{2}\bigr]$. The range is $\bigl[0,2\bigr]$. ...
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1answer
33 views

Finding the volume of the area between two curves when rotated about the y-axis

I keep going down a rabbit hole when answering this question: Find the volume of the area lying in the first quadrant and bounded by the $y$-axis, the curve $y = x^3$ and the line $y = 3x + 2$, ...
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1answer
41 views

How to calculate volume and surface area of three dimensional figures given set of three dimensional coordinates?

I have set of three dimensional coordinates, and the shape is unknown. I would like to calculate the surface area and volume for these coordinates approximately. What is the right approach to solve ...
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1answer
34 views

Find the max volume using polynomials with the sum of the height and perimeter less than 100cm

I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than 100cm. There is ...
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1answer
31 views

Math and volume of a liquid

I am trying to calculate the amount of chlorine in water- I have to dilute it because its hyper chlorinated and the testing system only goes up to 10 parts per million (chlorine should be 50ppm and ...
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2answers
48 views

Volume of a sphere “corner”

I would like to find the formula of the volume of the "corner" of a sphere of radius R, more specifically the volume delimited in a sphere by the intersection of two perpendicular planes, one parallel ...