Turtle Posted March 18, 2005 Report Share Posted March 18, 2005 ___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. Quote Link to comment Share on other sites More sharing options...
Qfwfq Posted March 21, 2005 Report Share Posted March 21, 2005 Have you tried cylindrical coordinates? Quote Link to comment Share on other sites More sharing options...
Turtle Posted March 21, 2005 Author Report Share Posted March 21, 2005 ___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. :friday: Quote Link to comment Share on other sites More sharing options...
Aki Posted March 22, 2005 Report Share Posted March 22, 2005 What is the question? Quote Link to comment Share on other sites More sharing options...
C1ay Posted March 22, 2005 Report Share Posted March 22, 2005 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. Quote Link to comment Share on other sites More sharing options...
Turtle Posted March 23, 2005 Author Report Share Posted March 23, 2005 ___Great wireframes C1ay! Surface area & volume aside for the moment, how do you think these shapes will behave when spun as tops? :friday: Quote Link to comment Share on other sites More sharing options...
Tim_Lou Posted March 23, 2005 Report Share Posted March 23, 2005 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 = 4second one= x^2 + (y+1)^2 = 4solve 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)-1area 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. Quote Link to comment Share on other sites More sharing options...
C1ay Posted March 23, 2005 Report Share Posted March 23, 2005 ___Great wireframes C1ay! Surface area & volume aside for the moment, how do you think these shapes will behave when spun as tops? :friday: 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. Quote Link to comment Share on other sites More sharing options...
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