Part II: Free Response. Solve each problem and show all work clearly. Draw a box around your final answers. Give exact symbolic answers and include units where applicable. There are $3$ problems worth a total of $44$ points.

$[14$ points$]$ Consider the double integral

${\displaystyle \underset{-2}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{4-x^2}}{}}}\sqrt[\phantom{2}]{x^2+y^2+4}dydx$

Sketch the region of integration, then evaluate the integral by switching to polar coordinates. Give an exact symbolic answer; no decimal approximations.