Step 1

Given integral

\(\displaystyle{\int_{{{0}}}^{{{2}}}}{\int_{{{0}}}^{{{1}}}}{4}{x}{y}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{4}{\left[{x}{\left.{d}{x}\right.}\right]}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{4}{\frac{{{x}^{{{2}}}}}{{{2}}}}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{2}{{\left[{x}^{{{2}}}\right]}_{{{0}}}^{{{1}}}}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{2}{\left[{1}^{{{2}}}-{0}^{{{2}}}\right]}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{2}{y}{\left.{d}{y}\right.}\)

Step 2

\(\displaystyle={2}{{\left[{\frac{{{y}^{{{2}}}}}{{{2}}}}\right]}_{{{0}}}^{{{2}}}}\)

\(\displaystyle={{\left[{y}^{{{2}}}\right]}_{{{0}}}^{{{2}}}}\)

\(\displaystyle={2}^{{{2}}}-{0}^{{{2}}}\)

=4

Therefore, the result of \(\displaystyle{\int_{{{0}}}^{{{2}}}}{\int_{{{0}}}^{{{1}}}}{4}{x}{y}{d}{x}{\left.{d}{y}\right.}\) is 4.

Given integral

\(\displaystyle{\int_{{{0}}}^{{{2}}}}{\int_{{{0}}}^{{{1}}}}{4}{x}{y}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{4}{\left[{x}{\left.{d}{x}\right.}\right]}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{4}{\frac{{{x}^{{{2}}}}}{{{2}}}}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{2}{{\left[{x}^{{{2}}}\right]}_{{{0}}}^{{{1}}}}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{2}{\left[{1}^{{{2}}}-{0}^{{{2}}}\right]}{y}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{{0}}}^{{{2}}}}{2}{y}{\left.{d}{y}\right.}\)

Step 2

\(\displaystyle={2}{{\left[{\frac{{{y}^{{{2}}}}}{{{2}}}}\right]}_{{{0}}}^{{{2}}}}\)

\(\displaystyle={{\left[{y}^{{{2}}}\right]}_{{{0}}}^{{{2}}}}\)

\(\displaystyle={2}^{{{2}}}-{0}^{{{2}}}\)

=4

Therefore, the result of \(\displaystyle{\int_{{{0}}}^{{{2}}}}{\int_{{{0}}}^{{{1}}}}{4}{x}{y}{d}{x}{\left.{d}{y}\right.}\) is 4.