$\int_D 1$, where $D:=\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = r^2, (r, z) \in C\}$. (Problem 3-29 in "Calculus on Manifolds" by Michael Spivak)

Question

The following problem is a problem in the section "FUBINI'S THEOREM".

Problem 3-29.
Use Fubini's theorem to derive an expression for the volume of a set of $\mathbb{R}^3$ obtained by revolving a Jordan-measurable set in the $yz$-plane about the $z$-axis.

Let $C$ be a Jordan-measurable set in the $yz$-plane.
Let $D:=\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = r^2, (r, z) \in C\}$.
We need to derive an expression for the volume of $D$ using Fubini's theorem.
This is our goal.

1. But first of all, we need to show $D$ is Jordan-measurable.
2. Next we need to derive a convenient expression for calculation of $\int_D 1=\int_E \chi_D$ using Fubini's theorem, where $E$ is a closed rectangle which contains $D$.

About 1.
How to show $D$ is Jordan-measurable?
I know we need to use the assumption that $C$ is Jordan-measurable but I have no idea.

About 2.
I also have no idea.

I would appreciate it if you could provide me with an outline of a solution or some hints.

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I know the following result. (But I don't know the proof.)

Pappus’s theorem, in mathematics, theorem named for the 4th-century Greek geometer Pappus of Alexandria that describes the volume of a solid, obtained by revolving a plane region $D$ about a line $L$ not intersecting $D$, as the product of the area of $D$ and the length of the circular path traversed by the centroid of $D$ during the revolution.

3. How to find the distance $d$ between the $z$-axis and the centroid of an arbitrary Jordan-measurable set $C$ in the $yz$-plane using the knowledge up to the section "FUBINI'S THEOREM"?

4. How to prove Pappus's theorem using the knowledge up to the section "FUBINI'S THEOREM"?

If we find the distance $d$, then the answer is $\left(\int_C 1\right)\cdot (2\pi d)$ by Pappus's theorem. (I assumed the $z$-axis doesn't intersect $C$.)