Leah Pope

2022-06-24

Calculate

${\int}_{\mathrm{\partial}F}2xy\phantom{\rule{thinmathspace}{0ex}}dx+{x}^{2}\phantom{\rule{thinmathspace}{0ex}}dy+(1+x-z)\phantom{\rule{thinmathspace}{0ex}}dz$

for the intersection of $z={x}^{2}+{y}^{2}$ and $2x+2y+z=7$. Go clockwise with respect to the origin.

${\int}_{\mathrm{\partial}F}2xy\phantom{\rule{thinmathspace}{0ex}}dx+{x}^{2}\phantom{\rule{thinmathspace}{0ex}}dy+(1+x-z)\phantom{\rule{thinmathspace}{0ex}}dz$

for the intersection of $z={x}^{2}+{y}^{2}$ and $2x+2y+z=7$. Go clockwise with respect to the origin.

Rebekah Zimmerman

Beginner2022-06-25Added 32 answers

You should consider the surface S given by the intersection of the cylinder $(x+1{)}^{2}+(y+1{)}^{2}\le 7+2=9$ with the plane $2x+2y+z=7$. Then $\mathbf{n}=-(2,2,1)/3$ (the ellipse $\mathrm{\partial}S$ is clockwise oriented) and by Stokes' theorem:

$\begin{array}{rl}{\int}_{\mathrm{\partial}S}\mathbf{F}\phantom{\rule{thinmathspace}{0ex}}d\mathbf{s}& ={\iint}_{S}\text{curl}(\mathbf{F})\phantom{\rule{thinmathspace}{0ex}}d\mathbf{S}\\ & ={\iint}_{(x+1{)}^{2}+(y+1{)}^{2}\le 9}(0,-1,0)\cdot (-(2,2,1))\phantom{\rule{thinmathspace}{0ex}}dxdy\\ & ={\iint}_{(x+1{)}^{2}+(y+1{)}^{2}\le 9}2\phantom{\rule{thinmathspace}{0ex}}dxdy\\ & =2\phantom{\rule{thinmathspace}{0ex}}\text{Area}(\{(x+1{)}^{2}+(y+1{)}^{2}\le 9\})=2\cdot 9\pi =18\pi .\end{array}$

$\begin{array}{rl}{\int}_{\mathrm{\partial}S}\mathbf{F}\phantom{\rule{thinmathspace}{0ex}}d\mathbf{s}& ={\iint}_{S}\text{curl}(\mathbf{F})\phantom{\rule{thinmathspace}{0ex}}d\mathbf{S}\\ & ={\iint}_{(x+1{)}^{2}+(y+1{)}^{2}\le 9}(0,-1,0)\cdot (-(2,2,1))\phantom{\rule{thinmathspace}{0ex}}dxdy\\ & ={\iint}_{(x+1{)}^{2}+(y+1{)}^{2}\le 9}2\phantom{\rule{thinmathspace}{0ex}}dxdy\\ & =2\phantom{\rule{thinmathspace}{0ex}}\text{Area}(\{(x+1{)}^{2}+(y+1{)}^{2}\le 9\})=2\cdot 9\pi =18\pi .\end{array}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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