How can we find the volume of an ellipsoid using an integral?

[-=nobody=-]

Well, I have a small problem. I know the general formula for the volume of an ellipsoid. But I have a task to find it with the help of an integral. Can you explain me how to do this?

 

[Curious3141]

We've covered this before. https://www.physicsforums.com/showthread.php?t=110799&highlight=volume+ellipsoid

And for the more special and simple case of a spheroid : https://www.physicsforums.com/showthread.php?t=76495&highlight=revolution

 

[-=nobody=-]

Thank you very much, the information is great.
And can you write the formula like in https://www.physicsforums.com/showpost.php?p=577097&postcount=12" but for an ellipsoid where a, b and c are different.

 

[siddharth]

As Curious3141 already posted, in the link below, HallsofIvy explains it very well. What part do you not understand?

https://www.physicsforums.com/showthread.php?t=110799&highlight=volume+ellipsoid"

 

[Curious3141]

Thank you very much, the information is great.
And can you write the formula like in https://www.physicsforums.com/showpost.php?p=577097&postcount=12" but for an ellipsoid where a, b and c are different.

No, that's for a spheroid (two axes equal). For the general ellipsoid use the triple integral method. Of course the final answer comes out to a simple $$V = \frac{4}{3}\pi abc$$, it's just the derivation that's involved.

 

[-=nobody=-]

And can you please show me how we can receive this
$$V = \frac{4}{3}\pi abc$$
from this
$$2c\int_{x=-a}^a\int_{y=-b\sqrt{1-\frac{x^2}{a^2}}}^{b\sqrt{1-\frac{x^2}{a^2}}}\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}dydx$$

And can we also use http://libraryofmath.com/math/Calcu...ple_Calculus_III_Volume_of_an_Ellipsoid.html" method?