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5.9 Examples of repeated integrals

Let $R$ be the rectangle $0\leq x\leq 2$ , $0\leq y\leq 1.$ Find the double integral

\int\!\!\!\int_{R}(x+y)\,dxdy.

Solution. We sketch the region, then calculate the repeated integral:

\int_{0}^{1}\int_{0}^{2}(x+y)\,dxdy=\,{\int_{0}^{1}\left[\frac{x^{2}}{2}+xy% \right]_{0}^{2}\,dy=}\,

{\int_{0}^{1}(2+2y)\,dy=}\,{\left[2y+y^{2}\right]_{0}^{1}=}\,{3.}