We'll probably need calculus here...

The axis of revolution isn't perpendicular to the region in question, so using the method of shells/disks can be really hard. Using the first Pappus-Guldinus theorem can allow us to compute the volume in terms of $a$ and $b$ easily.

The centroid of $R$ lies on the $y$ -axis (by symmetry), while the $y-$ coordinate is

$\bar{y} = \dfrac {\displaystyle \int_{0}^{b} y dA}{A} = \dfrac {\displaystyle \int_{0}^{b} y L(y) dy}{A}$

using the formula for the centroid of thin a plate ( $A$ is the area of the region). Since $x = \pm \frac {a}{b} \sqrt {b^2 - y^2}$ , we have $L(y) = \frac {2a}{b} \sqrt{b^2 - y^2}$ . Therefore,

$\bar{y} = \dfrac {\displaystyle \int_{0}^{b}\frac {2ay}{b} \sqrt {b^2 - y^2}}{\frac {1}{2}\pi ab} = \dfrac {\frac {-2a}{3b}(b^2 - y^2)^{3/2}\bigg |_{0}^{b}}{\frac {1}{2}\pi ab} = \frac {2ab^2}{3} \cdot \frac {2}{\pi ab} = \frac {4b}{3\pi}$

Therefore, the centroid lies at $\left(0, \dfrac {4b}{3\pi}\right)$ . From Pappus' theorem we know that

$V =2\pi A\rho = \pi^2 ab \rho = 3\sqrt{2}\pi^2 \rho$

where $\rho$ is the distance of the centroid from the axis of revolution. Hence, we need to find the distance between $\left(0, \dfrac {4b}{3\pi}\right)$ and the line $y = x-a$ . The equation of the line perpendicular to $y = x-a$ and passing through $\left(0, \dfrac {4b}{3\pi}\right)$ is $y = -x + \dfrac {4b}{3\pi}.$ These lines intersect at

$x-a = -x + \frac {4b}{3\pi} \Rightarrow x = \frac {a}{2} + \frac {2b}{3\pi} \Rightarrow \left(\frac {a}{2} + \frac {2b}{3\pi}, \frac {-a}{2} + \frac {2b}{3\pi}\right)$

$\therefore \rho =\sqrt {\left(\frac {a}{2} + \frac {2b}{3\pi}\right)^2 + \left(\frac {a}{2} + \frac {2b}{3\pi}\right)^2} = \frac {a}{\sqrt {2}} + \frac {2\sqrt{2}b}{3\pi} = \frac {3}{b} + \frac {2\sqrt{2}b}{3\pi}$

Therefore $V(b) = 3\sqrt{2}\pi^2(\frac {3}{b} + \frac {2\sqrt{2}b}{3\pi}).$ Minimizing by taking the derivative yields

$\frac {-3}{b^2} + \frac {2\sqrt{2}}{3\pi} = 0 \Rightarrow b_{\text {min}} = \frac {3}{2}\sqrt{\pi \sqrt{2}} \Rightarrow \lceil 1000b_{\text {min}} \rceil = \boxed {3162}$