How to find partial surface area of oblate spheroid?

[toomers]

Specifically, I would like to know the surface area that is cut by a plane coinciding with the z and x axes and at an equal distance throughout the remaining portion of the spheroid away from the cutting plane.

I already know the surface area of the complete oblate spheroid to be 2pi^2(1+((1-e^2)/e)atanh(e) where e=sqrt(1-c^2/a^2) conforming to the equation:

(x^2+y^2)/a^2 + z^2/c^2 = 1


From the derivation of the surface area, could someone please help me set up the integral expression to find only the surface area of a band that goes around the spheroid at the orientation mentioned?

Thanks!

 

[DrPhil]

Specifically, I would like to know the surface area that is cut by a plane coinciding with the z and x axes and at an equal distance throughout the remaining portion of the spheroid away from the cutting plane.

I already know the surface area of the complete oblate spheroid to be 2pi^2(1+((1-e^2)/e)atanh(e) where e=sqrt(1-c^2/a^2) conforming to the equation:

(x^2+y^2)/a^2 + z^2/c^2 = 1


From the derivation of the surface area, could someone please help me set up the integral expression to find only the surface area of a band that goes around the spheroid at the orientation mentioned?

Thanks!

I picture the globe, remembering that Earth is an oblate spheroid. Let the polar radius be c and let the equatorial radius be a > c. Cut the globe on the prime meridian, and look at the half containing Asia. The cut is an ellipse with semi-major axis c and semi-minor axis a. What is its circumference?

Use cylindrical coordinates with the cylinder axis in the y-direction: \(\displaystyle (y, \rho, \phi\)

Now make another plane parallel to the cut plane, at a distance y in the direction of Sumatra. That cut will also be an ellipse. Find the axes of that ellipse, a'(y) and c'(y), and express the ellipse as \(\displaystyle \rho(y, \phi)\).

 

[toomers]

I picture the globe, remembering that Earth is an oblate spheroid. Let the polar radius be c and let the equatorial radius be a > c. Cut the globe on the prime meridian, and look at the half containing Asia. The cut is an ellipse with semi-major axis c and semi-minor axis a. What is its circumference?

Use cylindrical coordinates with the cylinder axis in the y-direction: \(\displaystyle (y, \rho, \phi\)

Now make another plane parallel to the cut plane, at a distance y in the direction of Sumatra. That cut will also be an ellipse. Find the axes of that ellipse, a'(y) and c'(y), and express the ellipse as \(\displaystyle \rho(y, \phi)\).

I believe you accurately know what I am looking for. The ellipses bounded by the band will be different of course, but I seek to find the area of the band. I am unfamiliar with cylindrical coordinates.

If you see how the spheroid surface area is constructed, would this follow or that would simply be one of those 8 times the area of an octant?

I am hoping to get an equation from this.