a$.$ Show that ${\displaystyle \underset{0}{\overset{}{}}}{t}^n{e}^{-t}dt=n{\displaystyle \underset{0}{\overset{}{}}}{t}^{n-1}{e}^{-t}dt$ for all $n1.$

b$.$ Let $R$ be the region between the graph of $y=\sqrt[\phantom{2}]{x}{e}^{-ax}$ and the $x-$axis where $a$ is a positive constant. Let $V$ be the volume of the solid generated by revolving $R$ about the $x-$axis, and.let $W$ be the volume of the solid generated by revolving $R$ about the $y-$axis. Find all values of a for which $V=W.$

$[$In this question you may use the fact that ${\displaystyle \underset{0}{\overset{}{}}}\sqrt[\phantom{2}]{x}{e}^{-x}dx=\frac{\sqrt[\phantom{2}]{}}{2}.]$