The diagram displays a section of the curve $y = (x + 1)^2 + (x + 1)^{-1}$ together with the line $x = 1$. Point $A$ is the curve's minimum point.
(i)[5]
Demonstrate that the $x$-coordinate of $A$ obeys $2(x + 1)^3 = 1$, and determine the exact value of $\frac{d^2 y}{dx^2}$ at $A$.
(ii)[6]
Find, showing all necessary working, the volume formed when the shaded region is turned through $360^\circ$ about the $x$-axis.
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