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16 votes
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How to check if a sphere passes through another sphere when both travels in a straight line through 3d space

First of all, your problem is actually 1-dimensional, because the $y$ and $z$-coordinates of either ball will be zero at any time. As the balls are approaching each other, they will always intersect! ...
emacs drives me nuts's user avatar
11 votes

How to check if a sphere passes through another sphere when both travels in a straight line through 3d space

If the two spheres are centred at $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ at some time $t$, then the distance between their centres is given by the Euclidean distance $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)...
ConMan's user avatar
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9 votes
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On average, how many uniformly random real numbers from $0$ to $1$ are required for the sum of their squares to exceed $1$?

Simulation in R suggests something near $3.926$ is reasonable. ...
Henry's user avatar
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7 votes
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Why do these matrices within Pascal's triangle all have determinant $1$?

By definition on the lattice $\mathbb Z_+^2$, any entry $a_{ik}=a_{i-1,k} +a_{i-1,k+1}$ with all $a_{ik}=0,k>i$, $a_{1,1}=1$. It follows, that by subtraction of rows, without change of the ...
Roland F's user avatar
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6 votes

Calculating Volume of "fat sphere" defined by $x^8+y^8+z^8 = 64$

The technique is due to Dirichlet. It is spelled out fairly completely in the first edition of Whittaker and Watson, but reduced to a homework exercise by the fifth. Delighted to find that the 1902 ...
Will Jagy's user avatar
  • 141k
5 votes

Find the volume of a pyramid

We claim that the upper vertex of the pyramid is $V=(16,12,12)$ (and the face $CDE$ is orthogonal to the base $DEF$). It suffices to verify that the distance between $V$ and the vertices of the base, ...
Robert Z's user avatar
  • 146k
5 votes
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How do I express the region bounded by $z=x^2, z+y=1, z-y=1$ as a z-simple triple integral?

Not an answer. Just posting an image that hopefully makes it easier to grasp which region exactly is supposed to be covered. Normally I would add a picture to the question body, but this time I use a ...
5 votes
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Calculate the volume of intersection sphere and cone using triple integral

The solid has cylindrical symmetry. For $z \in [0, \sqrt{2}]$, you have a cone; for $z \in [\sqrt{2}, 2]$, you have a section of a sphere. Then $$V = \left[ \int_0^\sqrt{2} \int_0^{2 \pi} \int_0^z + \...
K. Jiang's user avatar
  • 8,304
5 votes

Finding the volume of a weird figure

First of all, it is impossible that the formula for a volume is in $\frac23 r^2$ for a reason of homogeneity. It must be $\frac23 r^3$. Proof : Up to a rotation that does not change the volume one can ...
Jean Marie's user avatar
  • 83.7k
4 votes

Calculating volume of a sphere using cross-sectional area

You are using the wrong formula for the area. The radius of the cross-section at height $h$ will be $r^2=R^2-h^2$ by the Pythagoras theorem. So the area is given by $A(h)=\pi r^2=\pi(R^2-h^2)$. ...
Ygor Arthur's user avatar
4 votes
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Non-compact manifolds with finite volume and conformal transformation

Yes, this is definitely possible. The idea is just to make a bunch of super tall spikes. Perhaps there are cleaner ways of “implementing” such spikes, but here’s one way. Since $M$ is non-compact, ...
peek-a-boo's user avatar
4 votes
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Finding a volume of region $0 \geq \cos(\theta_1 - \theta_2) + \cos(\theta_1 - \theta_3) + \cos(\theta_2 - \theta_3)$

First of all, three 3D representations of surface $(S)$ with implicit equation : $$\cos(x_1 - x_2) + \cos(x_1 - x_3) + \cos(x_2 - x_3)=0 \tag{1}$$ the first one in $(0, 2 \pi)^3$ with a main ...
Jean Marie's user avatar
  • 83.7k
4 votes

Calculating the Volume with three given expression

Without calculus... This is a trapezodial prism. The cross-section at $x=40$ is the trapezoid $y+z=140$, $20\leq y\leq 80$. Its area is $A_1=\tfrac{z_1+z_2}2(y_2-y_1)=\tfrac{120+60}2(80-20)=5400.$ ...
Bob Dobbs's user avatar
  • 11.5k
4 votes
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How is formula of the volume of trirectangular tetrahedron $\frac{A\times B\times C}{6}$?

Trirectangular tetrahedron is half of the cuboid No it's not. Here is a picture of the tetrahedron and the rest of the cuboid Even without calculation it should be clear that the tetrahedron is not ...
James K's user avatar
  • 1,310
4 votes
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Determinants & Volume

More properly, it should be $|\det A| \cdot V$. That said, think on a macro level, a very small parallelipiped (say, a cube $[0,\varepsilon]^3$ for a small $\varepsilon > 0$). Then the cube has ...
PrincessEev's user avatar
  • 45.2k
3 votes
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Volume of revolution of solid formed by $y=x^2$ and $y=2x$ about $y=-1$

tl;dr: Your work is correct; I just want to give an explanation that might help you be more confident in why. The way I like to think about these problems is in terms of integrating a cross-sectional ...
PrincessEev's user avatar
  • 45.2k
3 votes
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Finding the Volume of Region in Three-Dimensional Space.

The paraboloid $x^2 + y^2 \le 6-z$ intersects the plane $z=2$ $x^2+y^2 \le 4$ in a disk of radius 2. The image of the region in the plane $z = 2$ If we want to integrate in cartesian coordinates, we ...
user317176's user avatar
  • 11.5k
3 votes
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Calculating Volume of "fat sphere" defined by $x^8+y^8+z^8 = 64$

We consider computing the volume of the region $\mathcal{D}$ described by $$ \mathcal{D} = \left\{ (x, y, z) \in \mathbb{R}^3 : \left|\frac{x}{a}\right|^{\alpha} + \left|\frac{y}{b}\right|^{\beta} + \...
Sangchul Lee's user avatar
3 votes
Accepted

Evaluating triple integral under linear transformation

Let $f(x,y,z)=2x+y-2z$. Then\begin{align}\iiint_{T(C)}2x-y-2z\,\mathrm dx\,\mathrm dy\,\mathrm dz&=\iiint_{T(C)}f(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz\\&=\iiint_Cf\bigl(T(x,y,z)\bigr)|\...
José Carlos Santos's user avatar
3 votes
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What is the volume of this shape, and how can I calculate it?

The shape is symmetrical about the plane $x = 0.5$. From point $(1,0,0)$, there are two pyramid that when added make one (symmetrical) half of the solid. So the total volume will as follows $ V = 2 \...
Quadrics's user avatar
  • 23.9k
3 votes

Why can't I integrate with respect to x when using the shell method on a rotation about the x-axis?

If integrating with respect to $x$, then the method of washers is required, since the axis of revolution is the same as the direction of integration--both are with respect to $x$. So the differential ...
heropup's user avatar
  • 140k
3 votes
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Calculating the Volume with three given expression

You say : "I don't know multivariable calculus". This might be a good opportunity to learn a little bit. But to do this, you have to do things in order and make reminders about the single-...
Stéphane Jaouen's user avatar
3 votes
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A parallelepiped has a volume of $216 \text{ cm}^3$ and the total area $216 \text{ cm}^2$. Prove that the parallelepiped is a cube.

Let $ABCD-EFGH$ be the given parallelepiped which satisfies the followings : The volume is $216$, and the total area is $216$ $AB\parallel DC\parallel EF\parallel HG$ $AD\parallel BC\parallel EH\...
mathlove's user avatar
  • 143k
3 votes
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Find volume of body between surfaces

There are some issues: under the transformation $$x = \sqrt{2} r \cos \theta, \quad y = \sqrt{3} r \sin \theta, \tag{1}$$ your Jacobian is correct, and the limits of integration are correct, but the ...
heropup's user avatar
  • 140k
3 votes
Accepted

Finding the volume of a weird figure

First of all, notice that the plane intersecting the cylinder at $\frac{\pi}{4}$ means that the. plane can be expressed as $$z=x$$ You want to find the volume of the solid region using a double ...
user140776's user avatar
  • 1,953
2 votes

Calculate the volume of $G=\{(x,y,z) \in \Bbb R^3 : x^2+y^2+z^2 \leq 16 , 0 \leq z \leq 2 \}$

We denote by $E(z)$ the disc of radius $\sqrt{16-z^2}$. The volume of G is given by : $$\int_G dxdydz = \int_{z \in [0,2]} \left ( \int_{(x,y) \in E(z)} dxdy \right) dz $$ $$= \int_{z \in [0,2]}\int_{...
Zakaria 's user avatar
2 votes
Accepted

What is the relationship between the radius and height of water in the frustum of a right cone?

I think there is one missing piece of information: what is the angle between the axis of the frustum and the lines at its surface? Alternatively, what is the height of the "missing cone" at ...
Guillaume Geoffroy's user avatar
2 votes
Accepted

Why is the volume of some parallelepipeds given by the determinant?

The volume of a parallelepiped of vectors $\vec a=(a_1,a_2,a_3)$, $\vec b=(b_1,b_2,b_3)$, $\vec c=(c_1,c_2,c_3)$ is given as a triple product, like this: $$V=|\vec a\cdot(\vec b\times \vec c)|=\begin{...
bb_823's user avatar
  • 2,183
2 votes

What is the volume of the sphere in hyperbolic space?

It is easy to see that the volume of a sphere of radius $s$ in ${\mathbb H}^n$ (constant curvature -1) is given by $\operatorname{vol}(S^{n-1}) \sinh^{n-1}(s)$, where $S^{n-1}$ is a unit sphere in ...
Curtis McMullen's user avatar
2 votes
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Why can't we use discs with 'slanted edges' when calculating the volume of a solid of revolution?

I'm not going to be too careful in this post, just trying to get the idea across. The purple volume does not have volume equal to $\pi(R\cos\theta)^2\cdot R\delta\theta$. Its exact volume is hard to ...
FShrike's user avatar
  • 41.9k

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