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{\bf HW 10\\Math 261, F18}
This is the last homework set of the semester! This one is due at the beginning of your class on \underbar{\bf Friday, November 30}.
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\item Set up AND EVALUATE the integral(s) to compute the outward flux of the vector field $\mathbf F = \left$ for the region enclosed by the curves $y=x^2$ and $y=x$ in the first quadrant, \underbar{without} parameterizing the curves.
\item Use Green's Theorem (either version) to set up AND EVALUATE
$\displaystyle \int_C y^2dx+x^2dy$ where
$C$ is the triangle bounded by $x=0$, $x+y=1$, and $y=0$ (with counterclockwise orientation).
\item Parameterize the portion of the plane $2x-3y+z=5$ over the rectangle $0\leq x \leq 1$, $2\leq y \leq 5$. In particular, please provide $\mathbf r(x,y)$ in the parameters $x$ and $y$.
\item Suppose some surface $S$ is parameterized by $\mathbf r(u,v) = \langle u^2, uv, v^2\rangle$, for $-1\leq u\leq 1$, $0\leq v\leq 2$. Set up but DO NOT EVALUATE an integral to find the surface area of $S$.
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There will not be time to cover the divergence theorem before this final homework set is due, so here's one to try on your own time before the final exam (\underbar{not worth any points}):
Let $\textbf{F}=\left<3x^2,-2xy,-3xz\right>$ and let $D$ be the solid cut from the first octant by the plane $x+2y+z=2$. Write down an integral to find the outward flux of $\textbf{F}$ across the boundary of $D$ using the Divergence Theorem, i.e., set up (BUT DO NOT EVALUATE) the triple integral in the Divergence Theorem. Use the variable order $dz\ dy\ dx$.
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