How did I arrive at a logical contradiction?

Question

Assume we've defined a solid cylinder $ x^2+y^2\leq 2y$ It follows that $ x^2+(y-1)^2 \leq 1$. We have a solid cylinder of radius one. Now let $ y-1$ = $ \sin(\theta) $. Would it be a contradiction to say: by hypothesis, $ x^2+y^2 \leq 2\sin(\theta)+2$ because we can plug in our new y-value from $y-1=\sin(\theta)$? It naively seems valid to say $ x^2+y^2 \leq 2\sin(\theta)+2$ but we arrive at a contradiction because the cylinder has a radius of 1 and $2\sin(\theta)+2 \geq 1$ =radius of cylinder. I don't know what mistake I made to arrive at a contradiction.

Answer 1

Your cylinder base (the circle in the $xy$ plane) has a center in $(x,y)=(0,1)$ and just touches the $x$-axis. Thus, $\sqrt{x^2+y^2}$, which is the distance from the origin, does vary from $0$ (the bottom) to $2$ (the top), i.e. $x^2+y^2$ varies from $0$ to $4$.