Write an iterated integral for ${}_{R}1dA$ over the described region $R$ using $($a$)$ vertical cross$-$sections, $($b$)$ horizontal cross$-$sections.

Evaluate the following double integrals:

$($a$){\displaystyle \underset{1}{\overset{2}{}}}{\displaystyle \underset{y}{\overset{y^2}{}}}1dxdy$

$($b$){\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{1-s^2}}{}}}8tdtds$

$($c$){}_R\frac{y}{x^5+1}dA,$ where $R=\{(x,y)|0x1,0y{x}^2\}$