Find the Average Value Of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$

Multiple Choice

Find the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over the region R: square with vertices $( 0,0 ) , ( 2,0 ) , ( 2,2 ) , ( 0,2 )$ .

A) $\frac { 8 } { 3 }$
B) $\frac { 32 } { 3 }$
C) $\frac { 16 } { 5 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 8 } { 5 }$