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How to intuitively see that the $\text{volume of a pyramid }= 1/3 \times (\text{ area of base}) \times (\text{height})$

Here is a visual explanation::
John's user avatar
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Finding the volume of a weird figure

The area of a circle of radius $1$ is $\pi,$ which we can approximate by adding the areas of rectangles between chords of the circle. For half a circle, we can just use the area of half of each ...
David K's user avatar
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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
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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
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Areas versus volumes of revolution: why does the area require approximation by a cone?

That is because the volume approximation by the conical frusta is equivalent to that by the cylindrical disks in 3D, analogous to the area approximation by the trapeziums/trapezia is equivalent to ...
Pustam Raut's user avatar
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Calculate the volume section of sphere

From Mengchun Zhang comment's, I try answer my question as below. Given $r=2$ and $a=\sqrt{2}$, we have \begin{align*} &2hr=a^2+h^2\\ \iff & 4h=2+h^2\\ \iff & h^2-4h+2=0\\ \iff& h_{1,2}...
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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
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2 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
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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
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The volume of the parallelepiped

It seems like you know what is true regarding the relationship between determinants and parallelepiped volume. But you want to know why. Here's the way I like to think about it. The case of a diagonal ...
user469053's user avatar
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How to compute volume of a circle defined by L1 distance?

Outline of an inductive solution: Restrict the problem to cases where $r=\frac{1}{2}$ and $x$ is at the origin; it's easy to generalise to other cases. For any positive integer $i$, let $A_i$ be "...
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What is the volume of a semiellipsoid?

The equation has second degree in $(x,y,z)$. The cutting plane can be one of $(x-y,y-z,z-x)$, so due to an orthogonal mirror symmetry volume is divided into two equal parts. Symmetry is not disturbed ...
Narasimham's user avatar
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Volume-preserving vector fields in noncompact manifolds

We can sidestep the issue about the flow not being defined uniformly across points by treating the computation infinitesimally. For any manifold (compact or now) endowed with a volume form $\omega$, ...
Travis Willse's user avatar
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How to determine the volume of this ellipsoid with boundaries

We don't really need to evaluate a triple integral. A suitable scaling transformation $$(u,v,w) = (ax, by, cz)$$ transforms the ellipsoid to a unit sphere, and the plane $z = x$ is transformed to $w/...
heropup's user avatar
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