91Ó°ÊÓ

It is given that ${âˆ\neg }_{\mathrm{Î\copyright }}\left[f\right(x,y)+g(x,y\left)\right]dA={âˆ\neg }_{\mathrm{Î\copyright }}f(x,y)dA+{âˆ\neg }_{\mathrm{Î\copyright }}g(x,y)dA$

$R=\left\{\right(x,y)âˆ\pounds a≤x≤b\text{and}c≤y≤d\},\text{that is,}\mathrm{Î\copyright }⊆R\text{and}c$ is real number.

For any function$f$ that is continuous over region$\mathrm{Î\copyright }$

and $F\left(x,y\right)=f(x,y),\text{if}(x,y)∈\mathrm{Î\copyright }$and role="math" localid="1653944271832" $F\left(x,y\right)=0,\text{if}(x,y)∉\mathrm{Î\copyright }$

Writing as double Riemann sum, we get

${âˆ\neg }_R\left(F\right(x,y)+G(x,y\left)\right)dA=\underset{\mathrm{Δ}→0}{\lim }\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}\left[F\left(x_i^{*},y_j^{*}\right)+G\left(x_i^{*},y_j^{*}\right)\right]\mathrm{Δ}A$

and

$\mathrm{Δ}=\sqrt{(\mathrm{Δ}x{)}^2+(\mathrm{Δ}y{)}^2}$

Simplify RHS

${âˆ\neg }_R\left(F\right(x,y)+G(x,y\left)\right)dA=\underset{\mathrm{Δ}→0}{\lim }\left\{\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}F\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A+\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}G\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A\right\}$

$=\underset{\mathrm{Δ}→0}{\lim }\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}F\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A+\underset{\mathrm{Δ}→0}{\lim }\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}G\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A$

$={âˆ\neg }_{R}F(x,y)dA+{âˆ\neg }_{R}G(x,y)dA$

Using same property again

${âˆ\neg }_{\mathrm{Î\copyright }}\left[f\right(x,y)+g(x,y\left)\right]dA={âˆ\neg }_{\mathrm{Î\copyright }}f(x,y)dA+{âˆ\neg }_{\mathrm{Î\copyright }}g(x,y)dA$

The equation is true.

Changing order of sum

${âˆ\neg }_R\left(F\right(x,y)+G(x,y\left)\right)dA=\underset{\mathrm{Δ}→0}{\lim }\underset{j=1}{\overset{n}{∑}}\underset{i=1}{\overset{m}{∑}}\left[F\left(x_i^{*},y_j^{*}\right)+G\left(x_i^{*},y_j^{*}\right)\right]\mathrm{Δ}A$

Simplify RHS

${âˆ\neg }_R\left(F\right(x,y)+G(x,y\left)\right)dA=\underset{\mathrm{Δ}→0}{\lim }\left\{\underset{j=1}{\overset{n}{∑}}\underset{i=1}{\overset{m}{∑}}F\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A+\underset{j=1}{\overset{n}{∑}}\underset{i=1}{\overset{m}{∑}}G\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A\right\}$

$=\underset{\mathrm{Δ}→0}{\lim }\underset{j=1}{\overset{n}{∑}}\underset{j=1}{\overset{m}{∑}}F\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A+\underset{\mathrm{Δ}→0}{\lim }\underset{j=1}{\overset{n}{∑}}\underset{j=1}{\overset{m}{∑}}G\left(x_i^{*},y_j^{*}\right)\mathrm{Δ}A$

$={âˆ\neg }_{R}F(x,y)dA+{âˆ\neg }_{R}G(x,y)dA$

Using same property again

${âˆ\neg }_{\mathrm{Î\copyright }}\left[f\right(x,y)+g(x,y\left)\right]dA={âˆ\neg }_{\mathrm{Î\copyright }}f(x,y)dA+{âˆ\neg }_{\mathrm{Î\copyright }}g(x,y)dA$

Hence, equation is true.