Consider the double integral

${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{arcsiny}{\overset{\frac{}{2}}{}}}cos(x)\sqrt[\phantom{2}]{1+co{s}^{2}x}dxdy$

$(a)$ Sketch the region $in$ the $xy-$plane where the integral $is$ taken over.

$(b)$ Switch the order $of$ integration.

$(c)$ Compute the double integral.