Integration for the Volume of an Ellipsoid

[Daniel Sellers]

Homework Statement

Let E be the ellipsoid x^2 + 2xy +5y^ +4z^2 = 1
Find the Volume of E

Homework Equations

None, just various integration methods.

The Attempt at a Solution

I know we're not supposed to say 'I don't know where to start' but with this one I really don't. If the best approach is a coordinate change, how would I go about searching for the correct transformation to map this to a sphere?

I tried to solve a similar problem concerning an ellipse (not ellipsoid) and failed to find a substitution. Someone showed me how to solve that problem in cartesian coordinates but that method won't work here.

Can anyone with experience with this type of problem at least tell me whether it's best to search for a substitution or attempt the problem in cartesian? And what my first step in either direction ought to be?

 

[Mark44]

Homework Statement

Let E be the ellipsoid x^2 + 2xy +5y^ +4z^2 = 1
Find the Volume of E

Homework Equations

None, just various integration methods.

The Attempt at a Solution

I know we're not supposed to say 'I don't know where to start' but with this one I really don't. If the best approach is a coordinate change, how would I go about searching for the correct transformation to map this to a sphere?

I tried to solve a similar problem concerning an ellipse (not ellipsoid) and failed to find a substitution. Someone showed me how to solve that problem in cartesian coordinates but that method won't work here.

Can anyone with experience with this type of problem at least tell me whether it's best to search for a substitution or attempt the problem in cartesian? And what my first step in either direction ought to be?

Some advice I gave in another thread that wasn't helpful might actually be helpful here. If you rotate the x-y plane to a different, X-Y, coordinate system, it should be easier to find the points at which the ellipsoid intersects the various planes. The z-axis can stay unchanged -- just the x-y plane.

See https://en.wikibooks.org/wiki/Conic_Sections/Rotation_of_Axes.

Also, due to symmetry, you can find the volume in one octant of your transformed solid, then multiply that value by 8 to get the volume of the whold object.

 

[Daniel Sellers]

Some advice I gave in another thread that wasn't helpful might actually be helpful here. If you rotate the x-y plane to a different, X-Y, coordinate system, it should be easier to find the points at which the ellipsoid intersects the various planes. The z-axis can stay unchanged -- just the x-y plane.

See https://en.wikibooks.org/wiki/Conic_Sections/Rotation_of_Axes.

Also, due to symmetry, you can find the volume in one octant of your transformed solid, then multiply that value by 8 to get the volume of the whold object.

Unfortunately that doesn't seem to help, or maybe it would if I had a lot more practice with this type of problem. Applying that transformation results in a mess of trig functions which isn't any easier to interpret or integrate. This article doesn't explain how to find the angle I need to rotate through either.

I am trying to find a coordinate change which will map this ellipsoid to a sphere, but maybe that's not possible here?

 

[Daniel Sellers]

If I set z = 0 and solve for x in terms of y I can see that the major axis of the ellipse in the xy plane is the line x = -y which would need to rotated by pi/4 to align with a coordinate axis. Applying the transformation you recommended above and plugging in pi/4 results in 4X^2 + 2Y^2 + 4XY + 4z^2 which is slightly less ugly but still presents the same problem... and still seems to be a tilted ellipse.

If I use -pi/4 the numbers are a little different but the XY term is still there just with a -6 coefficient instead of 4

 

[Mark44]

This article doesn't explain how to find the angle I need to rotate through either.

Sure it does, in the first section under Graphing.
$\tan(2\theta) = \frac B {A - C}$. In your equation, A = 1, B = 2, and C = 5. When I put these numbers in I get a value for $\tan(2\theta)$ of -1/2.

I am trying to find a coordinate change which will map this ellipsoid to a sphere, but maybe that's not possible here?

You'll have to take care of the rotation first. Either way, I don't recommend turning the ellipsoid into a sphere.

If I set z = 0 and solve for x in terms of y I can see that the major axis of the ellipse in the xy plane is the line x = -y

I don't think so. That would be a rotation of -45°, which is too much.

If I set z = 0 and solve for x in terms of y I can see that the major axis of the ellipse in the xy plane is the line x = -y which would need to rotated by pi/4 to align with a coordinate axis. Applying the transformation you recommended above and plugging in pi/4 results in 4X^2 + 2Y^2 + 4XY + 4z^2 which is slightly less ugly but still presents the same problem... and still seems to be a tilted ellipse.

If you rotate by the right amount, the mixed term (xy term) goes away. The fact that you still have an XY term is a sign that your rotation angle isn't right.

 

[Daniel Sellers]

Sure it does, in the first section under Graphing.
$\tan(2\theta) = \frac B {A - C}$. In your equation, A = 1, B = 2, and C = 5. When I put these numbers in I get a value for $\tan(2\theta)$ of -1/2.

You'll have to take care of the rotation first. Either way, I don't recommend turning the ellipsoid into a sphere.

I don't think so. That would be a rotation of -45°, which is too much.

If you rotate by the right amount, the mixed term (xy term) goes away. The fact that you still have an XY term is a sign that your rotation angle isn't right.

Alright, that's helpful. I did ask for a starting place after all. The angle of rotation should be given by (1/2)Arctan(-1/2) then, correct? Am I going to need to compute a Jacobian for this transformation?

 

[Mark44]

Alright, that's helpful. I did ask for a starting place after all. The angle of rotation should be given by (1/2)Arctan(-1/2) then, correct? Am I going to need to compute a Jacobian for this transformation?

I don't think so, because it's only a rotation -- the lengths of things aren't going to change. I'm not positive in saying that the Jacobian isn't needed, though.
The Jacobian does play a role in converting from Cartesian to polar form -- that's where the r comes from in going from $dxdy$ to $rdrd\theta$.

 

[Daniel Sellers]

I don't think so, because it's only a rotation -- the lengths of things aren't going to change. I'm not positive in saying that the Jacobian isn't needed, though.
The Jacobian does play a role in converting from Cartesian to polar form -- that's where the r comes from in going from $dxdy$ to $rdrd\theta$.

You, sir, are a gentleman and a scholar. By applying that rotation I was able to eliminate the XY term and then do a simple (u,v,w) transformation, calculate it's Jacobian and then perform a simple integration over a sphere of radius 1. It worked out to a tidy pi/3 which was correct.

If I had to integrate a function over this volume, and hence was unable to perform a rotation, I still have no idea how I would do that. Thankfully that seems to be slightly beyond the scope of my class as we have moved onto to vector fields and gradients.

Thanks a ton

EDIT: Actually do know that I would have to find a direct substitution like I was trying to in the first place, but I don't know how to do THAT so, six of one...