Math CentralQuandaries & Queries


Question from Rahul:

I am not able to visualize a solid given by inequalities as under (in cylindrical coordinate system)
0=<r=<R, 0=<theta=<2pi, 0=<z=<r. It is said that it is a cylinder from which a cone is removed.
I know that (can visualize) first 2 inequalities say it is a cylinder of infinite length.
It seems that third inequality poses the restriction to the height as well as it makes us to remove a cone.
But I just can not understand it clearly.
Thanks in advance!

Hi Rahul,

Fix $\theta = 0$ and look at the $r-z$ plane identified by $\theta = 0.$ $R$ is a constant and the variable $r$ can take any value with $0 \leq r \leq R.$

theta=0 plane

Now let $r_0$ satisfy $0 \leq r_0 \leq R.$ The points $(r_0, \theta, z)$ is to satisfy the conditions in the problem are the points $0 \leq z \leq r_0,$ that is the points on the red line segment in my diagram.


If this vertical line segment is drawn for each value of $r$ with $0 \leq r \leq R$ you get the red triangle in the diagram below.


This is the region described in cylindrical coordinates by $0 \leq r \leq R, \theta=0, 0 \leq z \leq r.$ But in your question $\theta$ can take any value between $0$ and $2\;\pi.$ To construct this region you need to rotate this triangle around the z-axis to obtain a cylinder of radius $R,$ height $R$ with a conical section removed.


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