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

yogi55hr 2021-11-08 Answered
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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Expert Answer

Xyle1991
Answered 2021-11-09 Author has 9098 answers
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.
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