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Questions tagged [volume]

For questions related to volume, the amount of space that a substance or object occupies.

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Finding theta when using polar coordinates in solving double and triple integrals

So I have a problem with finding theta. I can always see from where to where the theta is going. The problem occurs when calculating volumes. Sometimes in the solutions book, for the full circle $$4\...
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0answers
28 views

Volume of $x^2 + 2y^2 + z^6 = 16$ [on hold]

How to calculate the volume enclosed by the area of the function below $x^2 + 2y^2 + z^6 = 16$ Also how can you draw such a function?
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21 views

What is the volume of a parallelotope knowing its heights?

For a $d$ dimensional parallelotope, if all of its heights $h_1, \ldots, h_d$ are known, how do I express its volume?
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What “cross products” do I need to find the volume of a cuboid?

In two dimensions, one can find the area for a quadrilateral by calculating two "cross products". If the vertices of the quadrilateral are $a, b, c, d$ clock-wise, consider the vectors $A = \vec{ab}$,...
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5answers
125 views

If $ \lim n\tan{\frac{180°}{n}}= \pi $, is $ \infty \times0 = \pi $?

I derived the volume of a cone using two approaches and compared the results. First I integrated a circle of radius $r$ over the height $h$ to get the expression: $$V_1=\frac{1}{3}\pi r^2 h$$ Then I ...
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1answer
37 views

Volume of a truncated elliptical cone with 2 different elliptical semi-axis on each end [closed]

I need to calculate the volume of a shape which is kind of like a truncated cone but has an elliptical base eg. Minor axis a: 5 Major axis b: 15 and a second more round elliptical top eg. Minor ...
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2answers
143 views

Understanding integration over Orthogonal Group

Let $M$ be an $n \times n$ matrix, and $O_n$ be the orthogonal group of $n \times n$ matrix. Calculate $ m_1 = \int_{O_n}tr(M)dV$ and $m_2 = \int_{O_n} tr(M)^2dV$ where tr(M) is defined as the trace ...
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29 views

Optimization of a Tetrahedron

What is the minimum surface area of a tetrahedron with a volume of 415.5 cm^3. Showing the optimization process would be appreciated.
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1answer
38 views

Computing to volume of ball intersection with cone and another ball

I am trying to calculate the integral $\ \int\int\int_V z dV $ where $\ V $ is the volume inside the ball $\ x^2 + y^2 + (z-2)^2 = 4 $ and also inside the cone $\ z^2 = x^2 + y^2 $ and outside the ...
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4answers
551 views

Computing volume inside a ball and outside a cylinder

I try to calculate the volume inside the ball $\ x^2 + y^2 + z ^2 = 4 $ the outside the cylinder $\ x^2+y^2=2x $ using double integral. Since the shape is symmetric I chose $\ z = 0 $ as bottom limit ...
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1answer
22 views

What is the volume of the part of the equation $(z+3)^2+y^2+x^2=25$ that lies above the $xy$-plane?

How do I find the volume of the part of the equation $(z+3)^2+y^2+x^2=25$ that lies above the $xy$-plane? I know that you have to use double (iterated) integrals and polar coordinates.
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1answer
51 views

Find volume between two surfaces with double integrals

Find the volume interior to $y^2+z^2=2$ and exterior to $x^2-y^2-z^2=2$. I have solved a few exercises of this kind, but all of them had one dependent function of $x$ and $y$ only, but here both of ...
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1answer
26 views

Volume of Solid through Integration

The solid shown below has a semicircular base of 2cm. Vertical cross-sections of the solid perpendicular to the diameter of the semicircle are right-angled triangles, the heights of which are bounded ...
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1answer
27 views

Finding a volume using double integrals

I am trying to find the volume of the region $$\{ (x,y,z) \in (\mathbb R_0^+)^3 \mid y^2+z^2 \leq 9 \land y^2 \geq 3x \}$$ The answer is $\frac{27}{16} \pi$. The volume should be able to find using ...
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0answers
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Finding volume of the solid bounded by graphs using double integral

My answer is wrong and I don’t what made it wrong. Given this equations and limits: $z=x^2, x=0, x=2, y=0, y=4$ It looks like a parabola that has a plan attached to it on its right leg. As I have set ...
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1answer
30 views

Calculating the area of the intersection between $S: x^2+y^2+z^2=4$ and $z\ge1$.

I started by drawing both graphs and found that the intersection is just the part of the sphere above $z=1$. So it's the part of the sphere from $1\le z\le2$ and let this be called $S^1$. I then let $...
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0answers
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Compute the volume of $ E(R,r)=\left\{ (x_1,x_2,x_3,x_4)\in\mathbb R^4:\frac{x_1^2+x_2^2+x_3^2}{R^2}+\frac{x_4^2}{r^2}\le 1 \right\}$

For positive real numbers $R$ and $r$, let $$ E(R,r)=\left\{ (x_1,x_2,x_3,x_4)\in\mathbb R^4:\frac{x_1^2+x_2^2+x_3^2}{R^2}+\frac{x_4^2}{r^2}\le 1 \right\}. $$ Find the volume of $E(R,r)$ by ...
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1answer
33 views

Volume integral over circle not around the origin

For an assignment I am asked to find the volume of a given volume R, namely $R=\left\{(x,y,z):0\leq z\leq\sqrt{4-x^2-y^2},(x-1)^2+y^2\leq1\right\}$. I have attempted solving this using cylindrical ...
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3answers
269 views

Attempting to prove $\int_0^{\infty} \frac{1}{x} dx$ converges

I'm sure countless people has tried to prove this but this is how I came across this, although I do remember seeing this in the past. Background https://www.youtube.com/watch?v=vQ0himyDR2E I was ...
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2answers
44 views

Revolving the region $y=6-x^2$, $y=2x+3$ about x=5

To find the volume of the resulting solid we can either use rings or shells (parallel or perpendicular slices) For this particular example using shells (parallel slices) makes it simple: We have $\int ...
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0answers
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Find the volume of a solid

Consider the solid bounded by 4$x^2$+$y^2$+$z^2$ = 9 and z$\ge$ $\sqrt{4x^2+y^2}$ Hint: Use and apply symmetry. Find the volume of the solid. I tried to convert the equations to ...
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1answer
26 views

Volume of a defined region using triple integrals

I was doing some excercises and I came upon this one, but I couldn't define the limits of integration. The problem says the following: Find the volume of the region defined by: $$z = x^2 + 3y^2 ~...
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1answer
17 views

How can I find the volume generated by revolving the following region about $x=5$?

The region enclosed by : $y=6-x^2$ and $y=5$ I first get the inverse functions and the intersections and then work with the disk/washer method, the result is zero and I can't figure out what am I ...
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1answer
24 views

Volume of a solid $y=\cos(x)$ and $y=0$ for the interval $0\le x \le \frac{\pi}2$

Volume of a solid $y=\cos(x)$ and $y=0$ for the interval $0\le x \le \frac{\pi}2$. I used method of shells to get: $$2\pi\int_{0}^{\pi/2}x\cos(x)\,dx.$$ And I got: \begin{align} 2\pi\int_{0}^{\...
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1answer
28 views

Volume of surface Revolution by area bounded by $y=\sin(x)$ and $y=0$

The revolution across the y-axis, and the bounded area is between $y=\sin(x)$, and $y=0$ for ${0\le x \le \pi}$. I did: $$V_{shells}= 2 \pi \int_\limits{0}^\pi x\sin(x)dx=2\pi^2$$ I am trying to do ...
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2answers
26 views

Volume of the pyramid - how to find the coordinates of 4 vertices?

The pyramid is bounded by planes: $x=0$, $y=0$, $z=0$ and $9x-y-3z=54$. IT is needed to calculate the volume $V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the ...
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0answers
35 views

Triple integral to calculate the volume of pyramid vs it's formula

It is well known the formula to calculate the volume of a pyramid: $V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the base to the apex. However I need to ...
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1answer
68 views

Maximum volume cylinder from a sheet of paper

We have an A4 sheet of paper and we have to build a cylinder of maximum volume using this paper by cutting out a rectangle and the two base circles in order to make the construction. My approach: ...
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1answer
37 views

Use washer method to find volume of region

I don't know how to find the outer and inner radius, of the given section bound by $y=-x^2+2x$ about y=-1 , in this case, the transversal section is about $x\in[0,2]$ by washed method. I know that $A(...
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1answer
26 views

The region between curves $y= {\sqrt x}$ , $0≤x≤4,y=1,x=4$ is revolved about $y=1$. Find the volume of a generated solid.

The region between curves $y= {\sqrt x}$ , $0≤x≤4,y=1,x=4$ is revolved about $y=1$. Find the volume of a generated solid. I believe I need to find volume outlined by green, blue and dotted, black ...
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1answer
53 views

How can we calculate the volume of a BCC Wigner-Seitz Cell? (Based on a imaginary cube)

Hello. As you can see in the picture, there’s this shape and shape’s surface consists of 6 square and 8 hexagon parts and I would like to know its volume but I don’t know where to start. The only ...
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2answers
29 views

Volume of Rotation of a Square

I have a square with vertices of $(0,5), (0,7), (-1, 6) $ and $ (1,6)$. This is to be revolved around the x axis to find the volume of rotation. How do I go about starting this problem? Thanks in ...
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1answer
47 views

Volume of Revolution about y=1

I have two functions and I am attempting to find the area enclosed by each when rotated around $y=1$. These said functions $y=x$ and $y=\sqrt{x}$. Upon graphing this it seemed simple however I am ...
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1answer
35 views

How to know if integration area is underneath or above curve. (Double integrals)

I am given the following integral: $\int \int_D f(x,y)dA =\int_0^1\left( \int_\sqrt{y}^1 f(x,y) dx\right)dy$ and I am asked to draw the integration area, $D$. I know that $D$ is the area ...
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0answers
18 views

Volume by rotation using polar coordinates

I have been trying to solve an exercise taken from the legendary brazilian book "Um Curso de Cálculo Volume 3" about change variable in double integral. This exercise literally says: Consider the ...
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1answer
36 views

I have no idea how to solve this problem using areas of known cross section

The problem involving cross sections I am so confused on how to find volume using known cross sections. I've never understood it. This problem that I've encountered is very difficult, and I tried ...
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2answers
25 views

Another formula for a cylinder's Volume??

Ok, so I thought about this but it doesn't make any sense. Let's say we have a cylinder with height h and the radius of the base r. Let's say I cut the cylinder vertically so it traces a rectangular ...
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0answers
16 views

Volume of a simplex: Specific case

I'm trying to see why for $x \in [0,1]$ the following holds: Let $v(x)$ represent the volume of the $(k-2)$-dimensional simplex $ \{ (X_1,\ldots,X_k):X_i > 0, X_2+ \cdots+X_k = 1-x\}$.Then $v(x) = ...
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0answers
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Find the volume of the region bounded by the planes

Find the volume of the region bounded by the planes: $$8x+7y+6z=5\\y=x,\quad x=0,\quad z=0$$ This is the integral I have set up but it's wrong $$\int_0^{5/8}\int_x^{\frac{1}{7}(5-8x)}\frac{1}{6}(5-...
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3answers
57 views

Volume of a truncated paraboloid

A body is surrounded by its lateral faces: $$z(x,y) = h \left(1 - \left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 \right)$$ and $$z(x,y)=0$$ It should be a paraboloid, right? How can I ...
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1answer
18 views

Lower Dimensional Volume under Transformation

It is a well known fact that if $K$ is a measurable set in $R^n$ (we can restrict to convex bodies if you like), and $T$ a linear transformation then $$|TK|=\left|\det T\right||K|.$$ If $K$ is not $n-$...
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1answer
45 views

Volume of a tilted cylinder

Suppose I have a tilted cylinder of length l inclined to the horizontal by an angle of $\theta$ then it's volume comes out to be same as that of a straight cylinder of height $l\sin\theta$. I tried to ...
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2answers
33 views

What is 3D volume of three vectors in $n$ Dimensional Space

We have three linearly independent vectors $v_1=(v_{11} , ... , v_{1n})$ , $v_2=(v_{21} , ... , v_{2n})$ , $v_3=(v_{31} , ... , v_{3n})$. We want to calculate the 3D volume of the Parallelepiped made ...
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0answers
25 views

Volume of intersection between a ball and a solid cylinder

Let $B_{R_1} \subseteq \mathbb{R}^3$ be the ball of radius $R_1 > 0$ centred at the origin. Let $C_{R_2,d}$ the solid cylinder with radius $R_2 > 0$ and with axis at distance $d$ from the ...
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1answer
23 views

Volume of a tent with canvas stretched from a circular base to a vertical semicircular rod

From George Simmons' Calculus With Analytical Geometry, page 229 question 7: A tent consists of canvas stretched from a circular base of radius $a$ to a vertical semicircular rod fastened to the ...
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4answers
71 views

Volume of a 3D simplex

I want to find the volume of the following simplex $$B := \{ (x,y,z) \in \mathbb{R^3} \mid x \geq 0, y \geq 0, z \geq 0, x+y+z \leq 2 \}$$ I tried to do it by evaluating a double integral but I'm ...
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0answers
40 views

How can I set this up? $\iiint dz\,dy\,dx$

Write an iterated triple integral in the order $dz\,dy\,dx$ for the volume of the region bounded below by the $xy$-plane and above by the paraboloid $z = x^2 + y^2$ and lying inside the cylinder $x^2 +...
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1answer
78 views

why is the pushforward of Haar on $SU(2)$ by trace the semicircle measure?

This is something which has been bugging me since I keep getting the wrong answer: that the pushforward measure should be the "square of the semicircle" rather than the semicircle measure. My ...
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1answer
18 views

Denoting a volume integral

Suppose I consider three dimensional vectors $x = (x_1,x_2,x_3) \in \mathbb{R}^3$ and I want to represent the volume $V$ comprising the vectors $x$ such that $f(x) \geq g(x)$, for some known functions ...
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1answer
39 views

Volume of a curve using integration [closed]

Consider the region $R$ bounded by the curves $y=ax^2+1, y=0, x=0,\space\text{and}\space x=1, \text{for}\space a\geq-1$. If $V_1(a)$ is the volume of the solid generated when $R$ is revolved about ...