Volume of cones


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The volume of a cone can be approximated by carving the cone into slices, and approximating each slice with a cylinder.

To then set it up as a Riemann sum, it helps to orient the cone so that we can sum along the -axis. Mathematica can assist with the visualization:

slicedCone

I don't know why there are grid lines on the slices; they do not appear in Mathematica.


n = 6; (* number of slices *)

h = 20; (* Height of cone *)
r = 15; (* Radius of base of cone *)

deltaX = h/n; (* thickness of each slice *)

x = 0;
boxes = List[];
For[i = 0, i < n, i++, x = x + deltaX; rAtX = x * r /h ; boxes = Append[boxes, Graphics3D[{Green, Opacity[0.1], EdgeForm[{Thin, Blue, Opacity[0.1]}], Cylinder[{{x - deltaX, 0, 0}, {x, 0, 0}}, rAtX]}]]]; Show[boxes, Boxed -> False, Axes -> True, AxesOrigin -> {0, 0, 0},
LabelStyle -> Directive[14]]

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