Pappus and Guldinus Theorems and Equations // How to Find the Surface Areas & Volumes of 3D objects

In this video I explain the theorems of Pappus and Guldinus and their equations. These equations help us to find the surface area of lines and curves rotated around an axis as well as the volumes of areas rotated around an axis.
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[@meteoriter1647]

Hi, thanks for the great video, I am trying to find the volume of a an entire tree, picture a silhouette of a tree, let's say a pine tree, I am thinking of a 21m high tree, the trunk is 0.4m diameter with a taper of the trunk at 0.8% on each side and the branches start at 1.8m off the ground and then there is a parabolic taper to the top of the tree from that point on to zero. The largest width of the tree, where the branches first start is 12m wide. I am thinking to revolve half the silhouette and so half of trapezoid of the trunk at 0.2m tapering to 1.8m high and then 6m branch start going parabolic to the top. Could you perhaps kindly outline how because I want to get this exact? Can this be done with an equation of the edge of the silhouette, or is it better to spin the area of half the silhouette, I'm not sure. Thank you.

[@studentengineering]

Sounds like an interesting problem. I think that your problem would be solved the easiest by just finding the cones and partial cones that make up the tree and add up those volumes. I don’t think many of the equations in the video need to be used because you are revolving the area in a full circle and the lines you are using are all straight.

[@meteoriter1647]

@@studentengineering I've done Pappus Guldinus theorem volume calculations in Calculus in high school, and wanted to do it once again, as the tree volumes, need to do a bulk density check for the entire enclosed volume of the trees, quite important because I have to analyze a parking garage roof slab with trees on top acting as additional loads on top of a thick soil overburden. FYI, the tree outline is sometimes parabolic and not always straight because there is a mix of maple/pine etc. Yes, I do understand that I could do cones and cylinders but preferred to do it in one shot. Not a problem but tks for the quick reply.