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4 votes
Accepted

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
1 vote

Volume-preserving vector fields in noncompact manifolds

We can also 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 $\...
Travis Willse's user avatar
1 vote
Accepted

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
  • 139k
1 vote
Accepted

Volume around $y$ axis

Below is an alternative approach, leading to the same result. You can first calculate the volume of the solid of revolution around $y$ bounded by $$y=x^2,\quad y=1,\quad x=0,$$ and then calculate the ...
HeroZhang001's user avatar
  • 2,536
1 vote

Shell and disc method applied to a specific integral

I strongly recommend that you read this answer first. After reading it, your question has $x = f(y) = 2/y$, between $y = 1$ and $y = 4$, revolved about the $y$-axis. Is the interval of integration ...
heropup's user avatar
  • 139k

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