Answer
$\overline{x}=\frac{1-k}{2-k}\cdot \frac{b^{2-k}-a^{2-k}}{b^{1-k}-a^{1-k}}$
Work Step by Step
$$N=\int_{a}^{b}Ax^{-k}dx$$
$$N=A[\frac{x^{1-k}}{1-k}]_{a}^{b}=A(\frac{b^{1-k}}{1-k}-\frac{a^{1-k}}{1-k})$$
$$\overline{x}=\frac{1}{N}A[\frac{x^{2-k}}{2-k}]_{a}^{b}=\frac{1}{N}A(\frac{b^{2-k}}{2-k}-\frac{a^{2-k}}{2-k})=\frac{1}{A(\frac{b^{1-k}}{1-k}-\frac{a^{1-k}}{1-k})}A(\frac{b^{2-k}}{2-k}-\frac{a^{2-k}}{2-k})=\frac{1-k}{2-k}\cdot \frac{b^{2-k}-a^{2-k}}{b^{1-k}-a^{1-k}}$$
