$1.$ For each shaded region pictured below:

$($i$)$ Write down a definite integral, or a sum of definite integrals, in the variable $\backslash ($ x $\backslash )($i$.$e$.$ slicing vertically$)$ that represents the area of the shaded region.

$($ii$)$ Write down a definite integral, or a sum of definite integrals, in the variable $\backslash ($ y $\backslash )($i$.$e slicing horizontally$)$ that represents the area of the shaded region.

$($b$)$

$($d$)$

$($c$)$

$($e$)$

$($f$)$