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\vtop{\hsize=3.32in \noindent {\bf Problem statement} A region $R$ in
${\bf R}^2$ is located in the first quadrant, as shown. Its boundary,
oriented counterclockwise as shown, is an interval $I=[2,5]$ on the
$x$-axis and a curve $C$ in the first quadrant.

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\noindent Suppose the following information is also known:

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\centerline{$\displaystyle\int\!\!\int_R 1\, dA = 5\,;\
\int\!\!\int_R x\, dA = 12\,;\
\int\!\!\int_R y\, dA = 8\,.$}


}

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\noindent Find $\displaystyle\int_C \left(x^2+xy+3y\right)dx+
\left(\arctan(y^3)+3x^2+2xy+x\right)dy$.

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\noindent {\bf Hint} $I+C$ is a positively (counterclockwise) oriented
piecewise smooth simple closed curve which is the boundary of $R$. Be
careful because the formulas for $P(x,y)=x^2+xy+3y$ and
$Q(x,y)=\arctan(y^3)+3x^2+2xy+x$ together have seven ``pieces''.



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