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

question 87

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.

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Definitions:

Deadweight Loss

The dip in economic performance due to the non-achievement or the impracticality of achieving the market balance for a good or service.

Government Programs

Initiatives and schemes developed and introduced by the government aimed at improving the welfare of its citizens, which can include healthcare, education, and social security among others.

Deadweight Loss

An economic inefficiency that occurs when the total amount of losses in welfare or surplus exceeds the gains, often due to taxes or monopolies.

Labor Taxes

Financial obligations placed on workers or their employers, calculated as a portion of the salaries paid to employees.