Question

Prove property\(II\).

Short answer

To prove\(LHS = RHS\).

Step-by-step solution

Proof:

Since,\(m \le f\left( {x,y} \right) \le m\)for all\(\left( {x,y} \right)\)in\(D\).

\(\begin{aligned}{l}\int {\int\limits_D {mdA} } \le \int {\int\limits_D {f\left( {x,y} \right)} } dA \le \int {\int\limits_D {mdA} } \\m\int {\int\limits_D {1dA} } \le \int {\int\limits_D {f\left( {x,y} \right)} } dA \le M\int {\int\limits_D 1 } dA\\mA\left( D \right) \le \int {\int\limits_D {f\left( {x,y} \right)} } dA \le mA\left( D \right)\end{aligned}\)