I have a conceptual question about parametrizing a 2-manifold in $\mathbf R^3$ and computing its area/volume. If there is anything that I didn't phrase correctly, I would be glad to change it.
What I understand: If we consider the sphere $S$ of radius $r$ in $\mathbf R^3$, then the intersection of $S$ with the plane $z=z_0$ is the circle
$$ x^2+y^2=r^2-z_0^2. $$
Now, let $A$ be the set of pairs $(t,z)$ where $t\in(0,2\pi)$ and $z\in(-a,a)$. If we define the function $\alpha:A\to\mathbf R^3$, where $$ \alpha(t,z)=\left(\sqrt{r^2-z^2}\cos t,\sqrt{r^2-z^2}\sin t,z\right), $$ then $\alpha$ is a chart for every point of $S$, except for the curve traced by the set of points of the form $\alpha(t,0)$. Since this set has measure zero, if we are interested in the area of $S$, it suffices to just compute the area for the points "covered" by $\alpha$: $$ \int_A\text dV=\int_Av(\text D\alpha)=\int_Ar=4\pi r^2. $$
What I'm having trouble on: Now let $\alpha(t),\beta(t),f(t):[0,1]\to\mathbf R$ be continuous functions. Then let $M$ be a 2-manifold in $\mathbf R^3$ whose intersection with the plane $z=t$ is the circle
$$ (x-\alpha(t))^2+(y-\beta(t))^2=(f(t))^2,\quad\text{whenever}\quad0\leq t\leq1, $$ and is empty otherwise. How can I go about parametrizing this manifold? Since $M$ is a 2-manifold, I know each point of $M$ should locally resemble $\mathbf R^2$. This suggests that I need to construct some sort of continuous function $g:A\to\mathbf R^3$ that parametrizes $M$ (or everything except a measure zero set of $M$). Would $g$ look something like $$ g(t,z)=(f(t)\cos t+\alpha(t),f(t)\sin t+\beta(t),\cdots), $$ or am I visualizing the definition of $M$ incorrectly? If I can define such a $g$, I think then $g$ would be a chart that works for all (or almost all) of $M$, so I can use the formula to find the area of $M$. I am told that if $\alpha$ and $\beta$ are constant and $f(t)=1+t^2$, then after setting up the integral I should be able to find the area of $M$. I'm also told that if $f$ is constant, $\alpha=0$, and $\beta(t)=at$, then the integral cannot be evaluated in terms of elementary functions.
Any hints on how to visualize $M$ or how to define the chart $g$ would be greatly appreciated!