smc

The graph of $x^2+y^2=4$ is a circle with radius $2$ centered at the origin. The graph of $y=|x|$ is the combined graphs of $y=x$ and $y=-x$ with a nonnegative y. Because the arguments of $y=x$ and $y=-x$ are $135^{\circ }$ and $45^{\circ }$ respectively, the angle between the graphs of $y=x$ and $y=-x$ is $90^{\circ }$. Thus, the smallest region bounded by the graphs is $\frac{1}{4}\cdot 2^2\cdot \pi =\boxed{\pi }$.