Calculate the volume of a tetrahedron whose vertices are $(1,1,1),(2,1,1),(1,2,1),(1,1,2)$.
I don't know how to start, do I have to find the planes?
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$\begingroup$ This is not a trivial problem. See here $\endgroup$– K.defaoiteAug 5, 2022 at 1:52
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1$\begingroup$ Finding the equations of the planes looks like a useful way to start this particular problem. $\endgroup$– David KAug 5, 2022 at 2:42
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1$\begingroup$ math.stackexchange.com/questions/1734604/… may help you. $\endgroup$– Gerry MyersonAug 5, 2022 at 2:43
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$\begingroup$ When trying to find a volume (or some other metric) of a region via integration, the bounds of the integrals come from the boundary equations. So yes, you should start by finding the equations of the bounding planes. (Fortunately they're not too hard in this case) $\endgroup$– AngelicaAug 5, 2022 at 2:44
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2$\begingroup$ math.stackexchange.com/questions/922879/… is almost exactly what you want. $\endgroup$– Gerry MyersonAug 5, 2022 at 2:44
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1 Answer
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Shift the tetrahedron by $(-1, -1, -1)$ then the vertices becomes
$(0,0,0), (1, 0, 0), (0, 1, 0), (0, 0, 1) $
So the volume is one third the area of the base times the height, which
$$ V = \frac{1}{3} \left( \dfrac{1}{2} (1)^2 \right) (1) = \dfrac{1}{6} $$
And we're done.
However, if you want to calculate the volume using integration, then you can use
$V = \displaystyle \int_{z=0}^1 A(z) \ dz $
where $A(z) $ is the cross sectional area of the tetrahedron at elevation $z$. Using similar triangles, or vector methods, one can show that the cross-section area is given by
$A(z) = A(0) (1 - z)^2 = \dfrac{1}{2} (1 - z)^2$
Hence, the volume is
$ V = \displaystyle \int_0^1 \dfrac{1}{2} (1 - z)^2 = \dfrac{1}{6} $
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$\begingroup$ That's good – but, given the tags, I suspect OP wants a solution using Calculus. $\endgroup$ Aug 5, 2022 at 2:39
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$\begingroup$ Do mention the scalar triple product ogf the edge vectors. $\endgroup$– Z AhmedAug 5, 2022 at 3:28
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$\begingroup$ good, but I was thinking dping it with a triple integration $\endgroup$ Aug 5, 2022 at 14:39
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2$\begingroup$ @HaitianSpaceman Details like that are important and should be part of the question text. You can edit the question to fix that. $\endgroup$– David KAug 5, 2022 at 16:38
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2$\begingroup$ Check the link in the comments above by @GerryMyerson $\endgroup$– Hosam HAug 5, 2022 at 16:41