Which has a larger area?

The yellow region is composed of semi circles. The diameter of each semicircle are the side lengths of the squares. We can used the formula $A=\dfrac{1}{2}\dfrac{\pi}{4}d^2$ where $d$ is the diameter. Note that we multiplied by $\dfrac{1}{2}$ because it is a semicircle. So the area of the yellow region is

$A_{yellow}=\dfrac{1}{2}\dfrac{\pi}{4}(5^2)(2)+\dfrac{1}{2}\dfrac{\pi}{4}(13^2)(2)+\dfrac{1}{2}\dfrac{\pi}{4}(22^2)=\dfrac{25}{4}\pi+\dfrac{169}{4}\pi+\dfrac{242}{4}\pi=109\pi \approx 342.43$

It follows that the area of the gray region is

$A_{gray}=\left[5^2-\dfrac{1}{2}\dfrac{\pi}{4}(5^2)\right](2)+(8^2)(2)+\left[11^2-\dfrac{1}{2}\dfrac{\pi}{4}(11^2)\right](2)=50-6.25\pi+128+242-30.25\pi=420-36.5\pi \approx 305.33$