$[$ACJC$15/$Promo$/$Q$12]$

The shaded region R in the diagram below is bounded by the curve $4(x+1{)}^2+(y-2{)}^2=4$ and the lines $y=-x$ and $x=-1.$

$($i$)$ Using the substitution $x=cos-1.$ show that

${\displaystyle \underset{-2}{\overset{-1}{}}}\sqrt[\phantom{2}]{-x^2-2x}dx=\frac{}{4}$

$($ii$)$ Hence or otherwise, find the exact area of $R.$

$[3]$

$($iii$)$ By translating both the curve and the line $y=-x$ one unit in the positive $x-$direction, or otherwise, find the volume of the solid formed when $R$ is rotated through $2$ radians about the line $x=-1.$

$[3]$

Ans: $($ii$)\frac{}{2}-\frac{1}{7}($iii$)3\mathrm{.}14$