= Solution
Neglect <gravitational focusing> and assume a fixed well-mixed volume $V$, $\alpha>1$, and $D_{\min}\ll D_{\rm cc}\ll D\ll D_{\max}$. For the small impactors that dominate the rate, the <geometric collision cross-section> is approximately $\pi D^2/4$. Thus
$$
R_{\rm cc}(D)\simeq\frac{\pi D^2}{4}\,v_pD^p\frac K V
\int_{D_{\rm cc}}^{D_{\max}}d^{-\alpha}\,dd.
$$
Evaluating and neglecting the upper-end term gives the <catastrophic planetesimal collision rate>
$$
\boxed{R_{\rm cc}(D)\simeq\frac{\pi Kv_p}{4V(\alpha-1)}D^{2+p}D_{\rm cc}^{1-\alpha}.}
$$
Using part (ii), its size dependence is
$$
\boxed{R_{\rm cc}(D)\propto D^{2+p+(1-\alpha)[1+(b-2p)/3]}.}
$$
If the destructive cutoff is near an endpoint or comparable to $D$, retain the full integral and the cross-section $\pi(D+d)^2/4$. The simple power law then need not apply.
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