Section 6.7 Surface Integral
6.7 Surface Integral
We have integration over a straight line, over a curve, over an flat region( [latex]xy[/latex]-plane), hence it is natural to talk about the integration over a surface. The application is finding the total heat or mass of a piece of thin surface. The idea is the same, we use Riemann sum.
Definition:
The surface integral of a scalar-valued function of [latex]f(x,y,z)[/latex] over a piecewise smooth surface [latex]S[/latex] having a vector equation [latex]\overrightarrow{r(u,v)}[/latex] is:
\[\int\int_{S}f(x,y,z)dS =\lim_{m,n\rightarrow\infty}\sum_{i=1}^{n}\sum_{j=1}^{m}f(P_{ij})\triangle S_{ij}\]
\[=\lim_{m,n\rightarrow\infty}\sum_{i=1}^{n}\sum_{j=1}^{m}f(P_{ij})|\overrightarrow{r_{u}}\times\overrightarrow{r_{v}}|\triangle u\triangle v\]
\[=\int\int_{D}f(\overrightarrow{r(u,v)})|\overrightarrow{r_{u}}\times\overrightarrow{r_{v}}|dudv.\]
Example 1: Find [latex]\int\int_{S}(x-y+z)dS[/latex] where [latex]S[/latex] has a vector equation [latex]\overrightarrow{r(u,v)}=\lt 2u-v,2u+v,2-u+v\gt[/latex] where [latex]0\leq u\leq1[/latex] and [latex]0\leq v\leq2[/latex].
Exercise 1: Find [latex]\int\int_{S}(x+y-z)dS[/latex] where [latex]S[/latex] has a vector equation [latex]\overrightarrow{r(u,v)}=\lt 1+u+v,u-2v,-u+v\gt[/latex] where [latex]0\leq u\leq2[/latex] and [latex]0\leq v\leq1[/latex].
Example 2: Find [latex]\int\int_{S}ydS[/latex] where [latex]S[/latex] has a vector equation [latex]\overrightarrow{r(u,v)}=\lt v,u\text{cos}(v),u\text{sin}(v)\gt[/latex] where [latex]1\leq u\leq2[/latex] and [latex]0\leq v\leq\pi[/latex].
Exercise 2: Find [latex]\int\int_{S}xdS[/latex] where [latex]S[/latex] has a vector equation [latex]\overrightarrow{r(u,v)}=\lt v\text{sin}(u),v\text{cos}(u),u\gt[/latex] where [latex]0\leq u\leq\frac{\pi}{2}[/latex] and [latex]0\leq v\leq1[/latex].
For a surface [latex]z=g(x,y)[/latex], we can define a parametric vector function using [latex]\overrightarrow{r(x,y)}=\lt x,y,g(x,y)\gt[/latex], hence we have [latex]|\overrightarrow{r_{x}}\times\overrightarrow{r_{y}}|=\sqrt{1+g_{x}^{2}+g_{y}^{2}}[/latex]
\[\int\int_{S}f(x,y,z)dS=\int\int_{D}f(x,y,g(x,y))\sqrt{1+g_{x}^{2}+g_{y}^{2}}dA.\]
Example 3: Find [latex]\int\int_{S}xdS[/latex] where [latex]S[/latex] is the part of the plane [latex]x+2y+z=6[/latex] lies in the first octant.
Exercise 3: Find [latex]\int\int_{S}ydS[/latex] where [latex]S[/latex] is the part of the plane [latex]2x+3y+z=12[/latex] lies in the first octant.
Example 4: Find [latex]\int\int_{S}xdS[/latex] where [latex]S[/latex] is the surface [latex]y=x^{2}+4z[/latex] where [latex]0\leq x\leq1[/latex] and [latex]0\leq z\leq1[/latex].
Exercise 4: Find [latex]\int\int_{S}zdS[/latex] where [latex]S[/latex] is the surface [latex]x=y+2z^{2}[/latex] where [latex]0\leq y\leq1[/latex] and [latex]0\leq z\leq1[/latex].
Example 5: Find [latex]\int\int_{S}(x-y-z)dS[/latex] where [latex]S[/latex] is the part of half cylinder [latex]y^{2}+z^{2}=4[/latex] with [latex]y\geq0[/latex] and lies between [latex]x=-1[/latex] and [latex]x=1[/latex].
Exercise 5: Find [latex]\int\int_{S}(x+y-z)dS[/latex] where [latex]S[/latex] is the part of half cylinder [latex]x^{2}+z^{2}=4[/latex] with [latex]x\geq0[/latex] and lies between [latex]y=0[/latex] and [latex]y=1[/latex].
Example 6: Find [latex]\int\int_{S}(xz)dS[/latex] where [latex]S[/latex] is the boundary enclosed by the cylinder [latex]y^{2}+z^{2}=4[/latex] and the planes [latex]x=0[/latex] and [latex]x=1[/latex].
Group work:
1. Find [latex]\int\int_{S}(yz)dS[/latex] where [latex]S[/latex] is the boundary enclosed by the cylinder [latex]x^{2}+y^{2}=4[/latex] and the planes [latex]z=0[/latex] and [latex]z=2[/latex].
2. Find [latex]\int\int_{S}(y)dS[/latex] where [latex]S[/latex] is the triangular region bounded by three vertices: [latex](1,0,0)[/latex], [latex](0,2,0)[/latex] and [latex](0,0,3)[/latex].
3. Find [latex]\int\int_{S}x^{2}dS[/latex] where [latex]S[/latex] has a vector equation [latex]\overrightarrow{r(u,v)}=\lt u-2v,u+v+3,u++2v\gt[/latex] where [latex]0\leq u\leq1[/latex] and [latex]0\leq v\leq1[/latex].