I was reading Zorich's about application of integral and come across the following. I would be very grateful if someone can answer my question.
We shall now prove a sufficient condition for an additive interval function to be generated by an integral, one that will be useful in what follows.
Proposition 1. Suppose the additive function $I(\alpha,\beta)$ defined for points $\alpha,\beta$ of a closed interval $[a,b]$ is such that there exists a function $f\in \mathcal{R}[a,b]$ connected with $I$ as follows: the relation $$\inf_{[\alpha,\beta]}f(x)(\beta-\alpha)\leq I(\alpha,\beta)\leq \sup_{[\alpha,\beta]}f(x)(\beta-\alpha)$$ holds for any closed interval $[\alpha,\beta]$ such that $a\leq \alpha\leq \beta\leq b$. Then $$I(a,b)=\int_{a}^{b}f(x)dx.$$
Then the author uses that proposition in order to find the volume of a solid of revolution.
Now suppose the curvilinear trapezoid shown in Fig. 6.2 is revolved about the closed interval $[a,b]$. Let us determine the volume of the solid that results. We denote by $V(\alpha,\beta)$ the volume of the solid obtained by revolving the curvilinear trapezoid $\alpha f(\alpha)f(\beta)\beta$ (see Fig.6.2) corresponding to the closed interval $[\alpha,\beta]\subset [a,b].$
According to our ideas about volume the following relations must hold: if $a\leq \alpha<\beta<\gamma\leq b,$ then $$V(\alpha,\gamma)=V(\alpha,\beta)+V(\beta,\gamma)$$ and $$\pi\left( \inf_{[\alpha,\beta]}f(x)\right)^2(\beta-\alpha)\leq V(\alpha,\beta)\leq \pi\left( \sup_{[\alpha,\beta]}f(x)\right)^2(\beta-\alpha).$$ In this last relation we have estimated the volume $V(\alpha,\beta)$ by the volumes inscribed and circumscribed cylinders and used the formula for the volume of a cylinder.
Then by Proposition 1 $$V(a,b)=\pi \int_a^b f^2(x)dx.$$
The last line bothers me a lot. To be honest I do not see how he applies Proposition 1. In order to apply Proposition 1 inequality should be $$ \inf_{[\alpha,\beta]}(\pi f^2(x))(\beta-\alpha)\leq V(\alpha,\beta)\leq \inf_{[\alpha,\beta]}(\pi f^2(x))(\beta-\alpha),$$ right? Can anyone explain to me that moment please?