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# Write the limits for the double integral$\int_R \int f(x, y) d A$for both orders of integration. Compute the area of $R$ by letting $f(x, y)=1$ and integrating. Triangle: vertices $(0,0),(3,0),(2,2)$

Solution

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$\int_{0}^{2}\int_{0}^{x}dydx+\int_{2}^{3}\int_{0}^{-2x+6}dydx=\int_{0}^{2}\int_{y}^{-.5y+3}dxdy$

To integrate in the order of $dydx$, the triangle must be broken up into 2 parts. Chop the triangle using $x=2$. The boundaries of the first triangle are $y=x, y=0, x=2$. To find out the upper $y$ limit in the second integral, find the equation of the line between (2,2) and (3,0). The boundaries of the second triangle are $y=-2x+6, y=0, x=2$. To integrate in the order of $dxdy$, you need to solve for $x$ in $y=x, y=-2x+6$.

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