. . . dang it. I am way disappointed Apprentice. And It's a really nice plot too. No, not at you but me: We wish to calculate the area of two elliptic cylinders. The red one and the blue one. The red one has height . . . red, and the blue one had height . . . uh, blue. The semi-major axes,

of both are 3, and the semi-minor axis,

is the green line in the new plot below which I've calculated to be

. Ok, assume that's all right for now. So we just need to calculate the areas of these two elliptic cylinders. Go now to Mathworld and get the formula for the area:

I'm not entirely sure of this but let's assume that's correct for now. So I calculated the red length is

. So the area of the red cylinder is:

which you can either calculate numerically like I did, or express the integral in terms of a complete elliptic integral of the second kind:

Now, suppose it was just a cylinder of height

and radius 3. Then the area would be

. Hey, that's close. If it were 1000 or so then I'd say who in here knows this better than me?. But that's close and gives me confidence I'm . . .uh, close. Can you do the blue one?