Deriving Expression for Volume of Ellipsoid

[tandoorichicken]

I've been asked to derive an expression for the volume of an ellipsoid. I know what the expression is, I just don't know how to get there from the information given. All that is given is that it is defined by

$$\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} \leq 1$$,

a standard inequality/definition for an ellisoid. Not quit sure how to go about this.

 

[HallsofIvy]

I've been asked to derive an expression for the volume of an ellipsoid. I know what the expression is, I just don't know how to get there from the information given. All that is given is that it is defined by

$$\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} \leq 1$$,

a standard inequality/definition for an ellisoid. Not quit sure how to go about this.

Have you tried anything? Like first drawing a picture. Certainly it is straight forward to set up the integral. The ellipsoid's widest girth comes in the xy-plane where it is an ellipse:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}= 1$$
That is is covered by taking x from -a to a and, for each x, y from $-b\sqrt{1-\frac{x^2}{a^2}$ to $b\sqrt{1-\frac{x^2}{a^2}}$. For each (x,y), z ranges from $-c\sqrt{1-\frac{x^2}{a^2}- \frac{y^2}{b^2}}$. The volume is
$$\int_{x=-a}^a\int_{y=-b\sqrt{1-\frac{x^2}{a^2}}}^{b\sqrt{1-\frac{x^2}{a^2}}}\int_{z=-c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}}^{c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}}dzdydx$$
or
$$2c\int_{x=-a}^a\int_{y=-b\sqrt{1-\frac{x^2}{a^2}}}^{b\sqrt{1-\frac{x^2}{a^2}}}\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}dydx$$

 

[arildno]

Use spherical coordinates, with a twist.

 

[Benny]

I agree with the previous poster, use spherical coordinates. You have (x²/a²) + (y²/b²) + (z²/c²) = 1. Spherical coordinates woudldn't work too well with that equation the way it is. It would be much better if your ellipse was transformed into a sphere with an equation like u^2 + v^2 + w^2 = 1. This suggests a change of variables. Consider u^2 = (x²/a²) and likewise for the other components for the three obvious substitutions.

 

[62646264]

if anyone could further evaluate this problem, i would greatly appreciate it. I followed the parameterization suggestion and set u=x/a, v=y/b, and w=z/c and switched the bounds on the integral. However, after integrating once I run into the problem of

2C * sqrt(1-u^2-v^2)

When I integrate that, I am unable to do it by hand and must do it on a computer system, which gives me a very long answer with two arctangents. I am pretty sure that will not work out to give me 4/3Piabc.

Thanks in advance.

 

[HallsofIvy]

That's not difficult to integrate. It is essentially $\sqrt{a^2- u^2}$ for which the standard substitution is the trig substitution $u= a sin(\theta)$. Here that would be $u= \sqrt{1- v^2}sin(\theta)$. Integrating that, with respect to u, will give something involving $\sqrt{1- v^2}$ which you do with another sin substitution.

 

[62646264]

I'm sorry but I don't understand. The formula listed in my previous post...

2C * sqrt(1-u^2-v^2)

is the resulting formula after having already integrated with respect to u.
You said "integrating that, with respect to u," but I need to integrate that with respect to v.

Also, if I introduce a sin(theta) into the equation, I don't see a point in the future at which that constant will disappear, and thus my answer would have a trig function in it, although the correct answer should be
4/3 * Pi * abc. Correct me if I'm wrong, I'm just not following your suggested method.

 

[Office_Shredder]

Ok, so just do 2C*sqrt[ (1-u^2)-v^2] and integrate with respect to v. By making the trig substitution, you're actually integrating with respect to sin(theta) instead, so it goes away then