Use the Midpoint Rule to Estimate $\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A$

Essay

Use the Midpoint Rule to estimate $\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A$ over $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 2 \}$ partitioned by the lines x = 1 and y = 1. Then estimate the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over R.