Volume of a Triangle Rotated about Different Lines

Question

Let's say I have a triangle formed bound by the x/y axes and the line $y=1-x$.
Say I rotate this triangle about the y-axis, forming a cone with $r=1$ and $h=1$. $v=\frac{1}{3} \pi r^2h$, so $v=\frac{\pi}{3}$ $u^3$.
Say I also rotate the original triangle about the line $x=1$, forming a "banked incline" whose volume is given by $v=\pi r^2h- \frac{1}{3} \pi r^2h$ or $v=\frac{2}{3} \pi r^2h$, so $v=\frac{2}{3} \pi$ $u^3$.
My question is this: why are the volumes of the solids different conceptually? I understand why this is true mathematically (for instance, the same problem can be done through integration), but lack the intuition behind this truth. The way I'm mistakenly thinking about this is that since both solids share the same cross-sectionional area no matter where you slice them, their volumes should also be identical, but clearly that's wrong.

Answer 1

Writing in Wolfram Mathematica 12.0:

cone1 = ParametricPlot3D[{u Cos[v], u Sin[v], 1 - u}, {u, 0, 1}, {v, 0, 2π},
                         Mesh -> None, PlotStyle -> Directive[Green, Opacity[0.2]]];

cone2 = ParametricPlot3D[{1 + u Cos[v], u Sin[v], u}, {u, 0, 1}, {v, 0, 2π},
                         Mesh -> None, PlotStyle -> Directive[Green, Opacity[0.2]]];

cylinder = ParametricPlot3D[{1 + Cos[v], Sin[v], u}, {u, 0, 1}, {v, 0, 2π},
                            Mesh -> None, PlotStyle -> Directive[Green, Opacity[0.2]]];

frames = Table[

      triangle1 = ParametricPlot3D[{u Cos[t], u Sin[t], v}, {u, 0, 1}, {v, 0, 1 - u},
                          Mesh -> None, PlotStyle -> Directive[Green, Opacity[0.8]]];

      triangle2 = ParametricPlot3D[{1 + u Cos[t], u Sin[t], v}, {u, 0, 1}, {v, 0, u},
                          Mesh -> None, PlotStyle -> Directive[Green, Opacity[0.8]]];

      binder1 = Show[{cone1, triangle1}, AxesLabel -> {x, y, z}, AxesOrigin -> {0, 0, 0},
                ImageSize -> {750, 750}, Boxed -> False, Method -> {"ShrinkWrap" -> True}, 
                PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}, ViewPoint -> {2.4, 2.2, 1.1}];

      binder2 = Show[{cone2, cylinder, triangle2}, AxesLabel -> {x, y, z}, 
                AxesOrigin -> {0, 0, 0}, ImageSize -> {750, 750}, Boxed -> False, 
                Method -> {"ShrinkWrap" -> True}, PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}, 
                ViewPoint -> {2.4, 2.2, 1.1}];

      GraphicsGrid[{{binder1, binder2}}], {t, 0, 2π, π/10}];

Export["rotation_solids.gif", frames, "AnimationRepetitions" -> ∞, "DisplayDuration" -> 1];

you can get the following gif image (by clicking on it, you can enlarge it):

from which it should be clear that:

• in the first case the volume of a cone ($V' = \pi/3$) is being brushed;
• in the second case the complementary volume of a cylinder ($V''=2\pi/3$) is being brushed.

In fact, $V'+V''= \pi\,$ which is the volume of a cylinder of unit radius and height.