Calculus Help Needed-Vesica Piscis Thread-Post#11

[Turtle]

___I have a calculus problem involving integration to find the volume & surface area of the figure traced in space by revolving the plane figure vesica piscis on its 2 axes. This is far as I can go with it, ie I know integration can do this, but I have no idea how to set it up or solve it.
___I wouldn't bother but I already have explored the plane geometry of the figure & I want to know more. Thanks!

The geometry & background on this investigation is in the thread 'Vesica Piscis--Real Sacred Geometry'; post#11 references the calculus I need.

[Qfwfq]

Have you tried cylindrical coordinates?

[Turtle]

___I looked in my calculus book & saw them using 3 methods, disks, washers, & cylinders I think. I simply don't understand it or know how to set it up. It's been many years since I took the course & even then I scraped through by the skin of my teeth. The calc book is now on the way to the recyclers.
___It may not be kosher, but I'm basically asking anybody who knows how to do this, to do it & post the answer is the vesica piscis thread. It's strctly a matter of my curiosity.

[Aki]

What is the question?

[C1ay]

Originally Posted by Aki

What is the question?

How does the surface area and volume compare if you bisect the vesica piscis on the x axis and revolve it on that axis versus the same on the y axis? Wireframe models attached below.

[Turtle]

___Great wireframes C1ay! Surface area & volume aside for the moment, how do you think these shapes will behave when spun as tops?

[Tim_Lou]

since the thingy involves with 2 circles of radius of 2...
lets set up equations of these 2 circles,
first one= x^2 + ( y-1)^2 = 4
second one= x^2 + (y+1)^2 = 4
solve for the system, and find the intersections.
so, the equation of the top curve of that "thing" would be the second circle, x^2 + (y+1)^2 = 4,
solve for y, y= sqrt (4 - x^2)-1
area would be the integral of pi*(y^2)dx---from (pi*r^2)--which is, pi*(sqrt (4 - x^2)-1)^2 dx using the intersections as bounds.

[C1ay]

Originally Posted by Turtle

___Great wireframes C1ay! Surface area & volume aside for the moment, how do you think these shapes will behave when spun as tops?

I believe the solid produced from bisecting and revolving about the x axis would certainly have more friction. I believe it would exhibit less precession than the solid produced using the y axis. Tops are effectively flywheels that store energy through angular momentum. Optimal flywheel usually have the bulk of their mass as far as is practical from their axis. I don't think either of the shapes derived from the vesica piscis would be optimal flywheels.