Volume of $\{(x,y,z)\in [a,b]\times \mathbb R^2:y^2+z^2\leq f(x)^2\}$ for a continuous function $f(x)$

Question

Let $f:[a,b]\rightarrow (0,\infty)$ be continuous. I want to calculate the Volume of $A=\{(x,y,z)\in [a,b]\times \mathbb R^2:y^2+z^2\leq f(x)^2\}$

So far I got this:

Using polar coodinates $(y=r\cos(\phi)$ and $y=r\sin(\phi)$) we have $$\int_A1=\int_0^{2\pi}\int_a^b\int_0^{f(x)}r\text{ }drdxd\phi.$$

My first question: It this correct? Secondly, if this is indeed correct I need to integrate $f(x)^2$. How can I do that?