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I have to find the volume of the ellipsoid described by the set $ E = \{(x,y,z) \in \mathbb{R}^3 |\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}<1\}$. I have a few ideas and there is a bit of literature regarding this problem (I'd like to solve it with triple integrals), but everything I've found uses the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. Is there a difference? Because to me it seems like the latter has no volume and is just describing a surface, with the implication that the volume that should be regarded is the volume enclosed by the surface I guess? Mainly I am concerned about wether it makes a difference in calculating the integral.

Edit: I should have added the sources I mentioned in the OP, here are some: https://en.wikipedia.org/wiki/Ellipsoid Volume of Ellipsoid using Triple Integrals What is the volume of an ellipsoid?

In all of them the Ellipsoid is described with an equal sign.

hzm
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  • Think in the case of the plane, what is $\frac{x^2}{a^2}+\frac{y^2}{b^2}(=1,<1)$?. Then, extend that idea to space. – A. P. Jan 03 '23 at 16:33
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    It would help to know what everything you've found consists of. – Lee Mosher Jan 03 '23 at 16:45
  • In the language of topology, say a closed $n$-ball $\mathbb{B}^n \in \mathbb{R}^n$, then $\partial \mathbb{B}^n$ is the boundary of the ball which is also known as $(n-1)$-sphere, $\partial \mathbb{B}^n=\mathbb{S}^{n-1}$. Now your open ellipsoid is topologically equivalent to an open ball $\mathbb{B}^3 \setminus \mathbb{S}^2$. – Ng Chung Tak Jan 03 '23 at 18:13

2 Answers2

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Not knowing what sources you have considered, I'll try to give a straightforward mathematical answer.

When evaluating the volume of a subset $X$ using an ordinary triple integral $\iiint_X dx \, dy \, dz$, it certainly does make a difference whether $X$ is a 2-dimensional surface such as $$S = \left\{(x,y,z) \in \mathbb R^2 \mid \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\right\} $$ or whether $X$ is, instead, a nonempty 3-dimensional open subset such as the region inside of $S$ given by $$E = \left\{(x,y,z) \in \mathbb R^2 \mid \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}<1\right\} $$ The point is that the 3-dimensional volume of a 2-dimensional surface such as $S$ is zero, whereas the 3-dimension volume of a 3-dimensional non empty open subset is nonzero: $$\iiint_S dx \, dy \, dz = 0 \quad\text{whereas}\quad \iiint_E dx \, dy \, dz \ne 0 $$

I'll leave it to you to continue on with the exact calculation of $\iiint_E dx \, dy \, dz$ if you desire, but a few more comments are in order. I suspect that your sources are not really very precise about the difference between the surface $S$ itself the region $E$ lying on the inside of $S$. One issue is that the rigorous computation of the mildly strange integral $\iiint_S dx \, dy \, dz = 0$ is not something usually considered in an ordinary multivariable calculus course. The ordinary multivariable Riemann integral is not really very well-defined for this purpose. Instead one would use the Jordan content or, even better, the multivariable Lebesgue integral.

Lee Mosher
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  • Apologies for not clarifying what my sources are.

    https://math.stackexchange.com/questions/3295400/what-is-the-volume-of-an-ellipsoid

    This is one example where the problem is tackled, there an equation is used.

    https://math.stackexchange.com/questions/2711676/volume-of-ellipsoid-using-triple-integrals Here is another with explicit use of the triple integrals.

    https://en.wikipedia.org/wiki/Ellipsoid

    This one is more subtle because technically there is no mention of how the volume is calculated, but at the beginning an equals sign is used aswell.

     – hzm Jan 03 '23 at 21:41
  • The first two links are indeed good examples of misuse of the equality sign, therefore leaving ambiguous whether one is taking the volume of the 2-dimensional surface or its 3-dimensional inside. But the wikipedia article is more careful. The first part of the article is entirely about the 2-dimensional surface. But then, under the heading Volume, you'll read that the first sentence says The volume *bounded* by the ellipsoid is... (emphasis added by me). – Lee Mosher Jan 04 '23 at 14:07
  • Ah, I must have missed that. The wiki does indeed justify it correctly. – hzm Jan 04 '23 at 18:20
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Let $E=\{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}\subset\Bbb{R^3}$ and $M=\{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}<1\}\subset\Bbb{R^3}$ be two subspaces of the $3$-dimensional real Euicledean space $\Bbb{R^3}$. We want to write down their differences as many as possible.

  1. Dimension: $\dim E=2$ and $\dim M=3$: Here, I mean dimension as manifolds. $E$ is a $2$-dimensional (closed) manifold. $E$ is locally homeomorphic to $\Bbb{R^2}$. But, $M$ is a $3$-dimensional manifold. $M$ is locally homeomorphic to $\Bbb{R^3}$.
  2. Opennes/Closedness: $E$ is a closed subspace whereas $M$ is open subspace of $\Bbb{R^3}$ with respect to the Euclidean topology.
  3. Homotopy/Homology groups: $M$ is a contractible space, so all homotopy/homology groups, except the zero-th ones, vanish. Whereas, $E$ is homeomorphic to the $2$-dimensional sphere $S^2$ and although it is simply connected, for example, $\pi_2(E)=\Bbb{Z}$, a non-trivial group. Similarly, $H_2(E)=\Bbb{Z}.$
  4. Differential geometric differences (This is connected with 1): $E$ is called a surface. It is parametrized by two coordinates. It has an area. I don't know how to call $M$... Open solid or object maybe. It has a volume. We should take its closure for the computation of the volume. Then, the surface area of $M$ is just area of $E$. There are more to say about their differential geometric differences.
Bob Dobbs
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