Question

WILL AWARD MEDAL
question attached

Answer 1

\[\int\limits_{}^{}\int\limits_{R}^{}\arctan(\frac{ y }{ x })\]

Answer 2

where R={(x,y)|1<=x^2+y^2<=4,0<=y<=x}

Answer 3

\[ f(x,y) = \arctan \frac{y}{x} = \arctan \frac {r \sin \theta}{r \cos \theta} = \arctan (\tan \theta) = \theta\]
just leaves
\[\int\limits_{\theta = 0}^{\pi/4}\int\limits_{r = 1}^{2} \theta\ r\ dr d \theta\]