Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads:

*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*.

By making some simplying assumptions about the plane region and changing variables to cylindrical coordinates we can obtain an expression for the volume. However, the material on the change of variables is treated two sections later (p. 66). Thus, **is there is a way to solve the problem without using change of variables?**

The definition of Jordan-measurable can be found on p. 56 (See also Theorem 3-9, p.55), but I sumarize it here: Spivak defines a bounded subset $C$ of $\mathbb{R}^{n}$ to be Jordan measurable if the topological boundary of $C$ has measure $0$, i.e. if for any $\varepsilon>0$ there is a cover $\{U\_{i}\}$ of $\mathrm{Bd}(C)$ by closed $n$-cubes such that $\sum\_{i}v(U\_{i})<\varepsilon$ (where $v(U\_{i})$ denotes the $n$-dimensional volume of the rectangle $U\_{i}$). If $C$ is Jordan measurable it is contained inside some closed $n$-cube $A$ and the characteristic function $\chi\_{C}$ is integrable on $A$. The integral $\int\_{A}\chi\_{C}$ is called the $n$-dimensional volume of $C$.