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Evaluate the integral below:
\n\\(\\int\\int_R\\left(\\var{a}x+\\var{b}y\\right)dxdy\\)
\nwhere \\(R\\) is the region of the plane enclosed by \\(\\var{a1}\\le \\var{r1}x+\\var{r2}y\\le \\var{b1}\\) and \\(\\var{a2}\\le \\var{r3}x+\\var{r4}y\\le \\var{b2}\\).
\n", "functions": {}, "advice": "\\(\\int\\int_R\\left(\\var{a}x+\\var{b}y\\right)dxdy\\)
\nwhere \\(R\\) is the region of the plane enclosed by \\(\\var{a1}\\le \\var{r1}x+\\var{r2}y\\le \\var{b1}\\) and \\(\\var{a2}\\le \\var{r3}x+\\var{r4}y\\le \\var{b2}\\).
\nLet \\(u=\\var{r1}x+\\var{r2}y\\) and let \\(v=\\var{r3}x+\\var{r4}y\\)
\nLimits:
\n\\(\\var{a1}\\le u\\le \\var{b1}\\) and \\(\\var{a2}\\le v\\le \\var{b2}\\)
\nJacobian:
\n\\(The\\,Jacobian\\,is\\,the\\,absolute\\,value\\,of\\,the\\,determinant:\\,\\,\\begin{vmatrix} \\frac{du}{dx}&\\frac{du}{dy}\\\\ \\frac{dv}{dx}&\\frac{dv}{dy}\\\\ \\end{vmatrix}=\\begin{vmatrix} \\var{r1}&\\var{r2}\\\\\\var{r3}&\\var{r4}\\\\\\end{vmatrix}=(\\var{r1})*(\\var{r4})-(\\var{r2})*(\\var{r3})=\\var{det}\\)
\n\\(dxdy=\\frac{1}{\\var{det}}dudv\\)
\n\\(\\implies \\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\left(\\var{a}x+\\var{b}y\\right)\\frac{1}{\\var{det}}dudv=\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}(\\simplify{{k}*{r1}}x+\\simplify{{k}*{r2}}y)\\,dudv=\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\var{k}(\\var{r1}x+\\var{r2}y)\\,dudv=\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\var{k}u\\,dudv\\)
\n\nInner integral:
\n\\(\\int_\\var{a1}^\\var{b1}\\var{k}u\\,du\\)
\n\\(=\\var{k}\\left(\\frac{u^2}{2}\\right)\\big|_\\var{a1}^\\var{b1}\\)
\n\\(=\\var{k}\\left(\\frac{(\\var{b1})^2}{2}\\right)-\\var{k}\\left(\\frac{(\\var{a1})^2}{2}\\right)\\)
\n\\(=\\var{inner}\\)
\n\nOuter integral:
\n\\(\\int_\\var{a2}^\\var{b2}\\var{inner}\\,dv\\)
\n\\(=\\var{inner}v\\big|_\\var{a2}^\\var{b2}\\)
\n\\(=\\var{inner}(\\var{b2})-\\var{inner}(\\var{a2})\\)
\n\\(=\\simplify{{inner}*({b2}-{a2})}\\)
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