Generalize the volume formula for Cylinder and Cone Frustum I'm trying to derive the Frustum Volume. Also I want to generalize the Formula to be applicable to the Cylinder, which is the case when r1 = r2.
Below picture is the typical derivation of it, and I know that the Formula holds whether r1 = r2 is true or not.
But there is a line in the proof that the devisor is r1 - r2. So it seems cheated if I say that this Formula can be used when r1 = r2.
So I want a guidance why this Formula holds even when r1 = r2. what is the reason behind that? I'm also not sure if this derivation is Mathematically correct and can be referred in my Thesis or not. or do you have any suggestion?
Frustum Volume

There is nothing wrong with this formula and it works for cylinder too.
The reason it works is that you can show that volume of frustum converges to the volume of cylinder as $r_2\to r_1$. Indeed, consider two frustums of the same height $h$. One with radii $r-h$ and $r$ and another with $r$ and $r+h$. Volume of cylinder with radius $r$ will be between volume of frustums. And their volumes converge to the same value as $h\to 0$
