91影视

It is given that

${鈭�}_{惟}cf(x,y)dA=c{鈭�}_{惟}f(x,y)dA$

Region is subset of rectangular region defined by

role="math" localid="1653943372914" $R=\left\{\right(x,y)鈭�a鈮�x鈮�b\text{and}c鈮�y鈮�d\}\text{, that is,}惟鈯�R\text{and}c$ is real number.

We know property of double integral

${鈭�}_{惟}f(x,y)dA={鈭�}_{R}F(x,y)dA$

and

localid="1653944299120" $F(x,y)=(x,y),\text{if}(x,y)鈭�惟$and $0,\text{if}(x,y)鈭�惟$

write the double integral on left hand side as double Reimann sum.

${鈭�}_{R}cF(x,y)dA=\underset{螖鈫�0}{\lim }\underset{i=1}{\overset{m}{鈭�}}\underset{j=1}{\overset{n}{鈭�}}cF\left(x_i^{*},y_j^{*}\right)螖A$and

$螖=\sqrt{(螖x{)}^2+(螖y{)}^2}$

Simplify RHS

${鈭�}_{R}cF(x,y)dA=c\underset{螖鈫�0}{\lim }\underset{i=1}{\overset{m}{鈭�}}\underset{j=1}{\overset{n}{鈭�}}F\left(x_i^{*},y_j^{*}\right)螖A$

$=c{鈭�}_{R}F(x,y)dA$

From same property ${鈭�}_{惟}cf(x,y)dA=c{鈭�}_{惟}f(x,y)dA$

Equation is true.

Changing order of sum

${鈭�}_{R}cF(x,y)dA=\underset{螖鈫�0}{\lim }\underset{j=1}{\overset{n}{鈭�}}\underset{i=1}{\overset{m}{鈭�}}cF\left(x_i^{*},y_j^{*}\right)螖A$

${鈭�}_{R}cF(x,y)dA=c\underset{螖鈫�0}{\lim }\underset{j=1}{\overset{n}{鈭�}}\underset{i=1}{\overset{m}{鈭�}}F\left(x_i^{*},y_j^{*}\right)螖A$

$=c{鈭�}_{R}F(x,y)dA$

From property stated above

${鈭�}_{惟}cf(x,y)dA=c{鈭�}_{惟}f(x,y)dA$