# Evaluate the following iterated integrals. \int_{0}^{2}\int_{0}^{1}4xy dx dy

Evaluate the following iterated integrals.
$$\displaystyle{\int_{{{0}}}^{{{2}}}}{\int_{{{0}}}^{{{1}}}}{4}{x}{y}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$

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Xyle1991
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.