Volumes 1

For $k>0$ , let $R_k$ be the region in $\mathbb{R}^3$ given by

$R_k = \{(x,y,z) : \left|x\right|^k + \left|y\right|^k \leq 1 \text{ AND } \left|y\right|^k + \left|z\right|^k \leq 1 \text{ AND } \left|z\right|^k + \left|x\right|^k \leq 1\}.$

Let $V_k$ be the volume of $R_k$ . Let $V_{\infty} = \lim_{k\rightarrow\infty}{V_k}$ . There exists integers $a,b,c$ such that

$\lim_{k\rightarrow\infty}{k^a (V_{\infty} - V_{k})} = b \pi^c.$

Submit $a+b+c+V_1+10 V_{1/2}+(V_2 - 16)^2$ .