91影视

An integral is given as${鈭�}_{R}xydA$

By definition the double integration is

${鈭�}_{R}f(x,y)dA=\underset{螖鈫�0}{\lim }\underset{j=1}{\overset{m}{鈭�}}\underset{k=1}{\overset{n}{鈭�}}f\left(x_j^{*},y_k^{*}\right)螖A=\underset{螖鈫�0}{\lim }\underset{k=1}{\overset{n}{鈭�}}\underset{j=1}{\overset{m}{鈭�}}f\left(x_j^{*},y_k^{*}\right)螖A$

where

$$\begin{gathered} x_j=a+j螖x \\ {y}_k=b+k螖y \\ 螖A=螖x脳螖y \\ 螖x=\frac{b-a}{m} \\ 螖y=\frac{d-c}{n} \end{gathered}$$

The starred points $\left(x_j^{*},y_k^{*}\right)$lets choose $\left(x_j,y_k\right)=(0+j螖x,1+k螖y)=(j螖x,1+k螖y)$ for each jand k

$$\begin{gathered} {鈭�}_{R}f(x,y)dA=\underset{螖鈫�0}{\lim }\underset{j=1}{\overset{m}{鈭�}}\underset{k=1}{\overset{n}{鈭�}}\left(j螖x\right)(1+k螖y)螖A \\ \underset{j=1}{\overset{m}{鈭�}}\underset{k=1}{\overset{n}{鈭�}}\left(j螖x\right)(1+k螖y)螖A=\underset{j=1}{\overset{m}{鈭�}}\left(j螖x\right)螖A\left(\underset{k=1}{\overset{n}{鈭�}}(1+k螖y)\right) \\ =\underset{j=1}{\overset{m}{鈭�}}\left(j螖x\right)螖A\left(\underset{k=1}{\overset{n}{鈭�}}1+\underset{k=1}{\overset{n}{鈭�}}k螖y\right) \\ =\underset{j=1}{\overset{m}{鈭�}}\left(j螖x\right)螖A\left(n+螖y\frac{n(n+1)}{2}\right) \\ =\left(n+螖y\frac{n(n+1)}{2}\right)螖A\left(\underset{j=1}{\overset{m}{鈭�}}\left(j螖x\right)\right) \\ =\left(n+螖y\frac{n(n+1)}{2}\right)螖A\left(螖x\underset{j=1}{\overset{m}{鈭�}}\left(j\right)\right) \\ =\left(n+螖y\frac{n(n+1)}{2}\right)\left(螖x\frac{m(m+1)}{2}\right)螖A \end{gathered}$$

Recall,

$$\begin{gathered} 螖x=\frac{b-a}{m}=\frac{2-0}{m}=\frac{2}{m} \\ 螖y=\frac{d-c}{n}=\frac{4-1}{n}=\frac{3}{n} \\ 螖A=螖x脳螖y=\frac{2}{m}脳\frac{3}{n}=\frac{6}{mn} \\ \underset{j=1}{\overset{m}{鈭�}}\underset{k=1}{\overset{n}{鈭�}}\left(j螖x\right)(1+k螖y)螖A=\left(n+螖y\frac{n(n+1)}{2}\right)\left(螖x\frac{m(m+1)}{2}\right)螖A \\ =\left(n+\frac{3}{n}\frac{n(n+1)}{2}\right)\left(\frac{2}{m}\frac{m(m+1)}{2}\right)\left(\frac{6}{mn}\right) \\ =6\left(n\frac{1}{n}+\frac{3}{n}\frac{1}{n}\frac{n(n+1)}{2}\right)\left(\frac{1}{m}\frac{2}{m}\frac{m(m+1)}{2}\right) \\ =6\left(1+\frac{3}{2}\frac{(n+1)}{n}\right)\left(\frac{(m+1)}{m}\right) \end{gathered}$$

The double integration is

$$\begin{gathered} {鈭�}_{R}f(x,y)dA=\underset{螖鈫�0}{\lim }\underset{j=1}{\overset{m}{鈭�}}\underset{k=1}{\overset{n}{鈭�}}\left(j螖x\right)(1+k螖y)螖A \\ =\underset{螖鈫�0}{\lim }6\left(1+\frac{3}{2}\frac{(n+1)}{n}\right)\left(\frac{(m+1)}{m}\right) \\ =\underset{m鈫�鈭�}{\lim }\left(\underset{n鈫�鈭�}{\lim }6\left(1+\frac{3}{2}\frac{(n+1)}{n}\right)\left(\frac{(m+1)}{m}\right)\right) \\ =6\underset{m鈫�鈭�}{\lim }\left(\underset{n鈫�鈭�}{\lim }\left(1+\frac{3}{2}\frac{(n+1)}{n}\right)\left(\frac{(m+1)}{m}\right)\right) \\ =6\underset{m鈫�鈭�}{\lim }\left(\underset{n鈫�鈭�}{\lim }1+\underset{n鈫�鈭�}{\lim }\left(\frac{3}{2}\frac{(n+1)}{n}\right)\left(\frac{(m+1)}{m}\right)\right) \\ =6\underset{m鈫�鈭�}{\lim }\left(\left(1+\frac{3}{2}\right)脳\left(\frac{(m+1)}{m}\right)\right) \\ =6\left(1+\frac{3}{2}\right)\left(\underset{m鈫�鈭�}{\lim }\frac{(m+1)}{m}\right) \\ =6\left(\frac{5}{2}\right)\left(1\right) \\ =15 \end{gathered}$$