\((2r-a)^2 + r^2 = (a+r)^2\)
\(4r^2 - 4ra +a^2 +r^2 = a^2 +2ra +r^2\)
\(4r^2 = 4ra +2ra = 6ra\)
\(a= \frac{2r}{3}\)
The area of the shade region is \(A_s = \pi a^2 \)
\(A_s = \pi (\frac{2r}{3})^2\)
\(A_s = \frac{4r^2}{9} \pi \) Answer