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Create solids of revolution and compute surface area and volume?

Ray Lawicki, COLLEGE
Posted 8 years ago

I am trying to get to plot a 2D Cartesian graph (no problem here) then I want to rotate this graph around an axis eg. x-axis to produce a 3D graph and ask mathematica to draw it and find its volume and surface area without me having to do the calculus.
POSTED BY: Ray Lawicki
6 Replies
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Ray Lawicki
Ray Lawicki, COLLEGE
Posted 8 years ago

i found your reply quite difficult to follow as i am just a novice with mathematica, however i have done the following Plot[{-4(x+1)(x-3),3(x+1)(x-3)},{x,-3,6}],Plot[repeat the above with a domain of {x,-1,18/7},Filling->{2->{1}] which draws what i want. now can mathematica calculate its area without calculus and what about revolving this area around the x-axis to show a volume and calculate the volume as a region again without calculus.
POSTED BY: Ray Lawicki

Then you need to formulate it as an ImplicitRegion

reg = ImplicitRegion[
   y^2 + z^2 <= x^4 && -2 <= x <= 2,
   {x, y, z}
   ];
RegionPlot3D[
 ImplicitRegion[
  y^2 + z^2 <= x^4 && -2 <= x <= 2,
  {x, y, z}
  ]
 ]
Volume[reg]
(*  (64 \[Pi])/5  *)

However, 

Area[reg]

does not seem to work. You can get an answer if you create a MeshRegion, but the answer is very wrong

reg2 = DiscretizeGraphics@Normal@RegionPlot3D@reg
Area@reg2
(* 206.267 *)
POSTED BY: Jason Biggs

Please stop using excessive capitalization of letters.

POSTED BY: Moderation Team
Ray Lawicki
Ray Lawicki, COLLEGE
Posted 8 years ago

THANKS FOR THE REPLY BUT WHAT I WANT IS TO PLOT A CARTESIAN EQUATION SUCH AS Y=X^2 (THAT I CAN DO) AND THEN ASK MATHEMATICA TO PLOT THE ROTATION OF THIS GRAPH ABOUT AN AXIS (SAY X-AXIS) AND THEN ASK MATHEMATICA TO CALCULATE ITS VOLUME(0<X<5)SAY AND ITS SURFACE AREA,LIKE Volume[region] and Area[region], where the region is specifed as an area bounde by the function, x axis and line x=5, can this be done without doing the integration explicitely.
POSTED BY: Ray Lawicki

For making a plot, you need to look at RevolutionPlot3D, read the docs and you should be able to make the plots.

If you want the area and volume, you could just evaluate 

WolframAlpha["rotate y=sin(x), 0<x<pi about the x-axis"]

or something similar
POSTED BY: Jason Biggs

old but on topic: Calculating volume between two surfaces of revolution
POSTED BY: Kuba Podkalicki
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