Section 6.7 Surface Integral

6.7 Surface Integral

We have integration over a straight line, over a curve, over an flat region( $xy$-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 $f(x,y,z)$ over a piecewise smooth surface $S$ having a vector equation $\overrightarrow{r(u,v)}$ is:

$$\int {\int }_{S}f(x,y,z)dS=\underset{m,n\to \mathrm{\infty }}{lim}\sum _{i=1}^n\sum _{j=1}^{m}f(P_{ij})\mathrm{△}{S}_{ij}$$

$$=\underset{m,n\to \mathrm{\infty }}{lim}\sum _{i=1}^n\sum _{j=1}^{m}f(P_{ij})|\overrightarrow{r_u}\times \overrightarrow{r_v}|\mathrm{△}u\mathrm{△}v$$

$$=\int {\int }_{D}f(\overrightarrow{r(u,v)})|\overrightarrow{r_u}\times \overrightarrow{r_v}|dudv.$$

Example 1: Find $\int {\int }_S(x-y+z)dS$ where $S$ has a vector equation $\overrightarrow{r(u,v)}=<2u-v,2u+v,2-u+v>$ where $0\le u\le 1$ and $0\le v\le 2$.

Exercise 1: Find $\int {\int }_S(x+y-z)dS$ where $S$ has a vector equation $\overrightarrow{r(u,v)}=<1+u+v,u-2v,-u+v>$ where $0\le u\le 2$ and $0\le v\le 1$.

Example 2: Find $\int {\int }_{S}ydS$ where $S$ has a vector equation $\overrightarrow{r(u,v)}=<v,u\text{cos}(v),u\text{sin}(v)>$ where $1\le u\le 2$ and $0\le v\le \pi$.

Exercise 2: Find $\int {\int }_{S}xdS$ where $S$ has a vector equation $\overrightarrow{r(u,v)}=<v\text{sin}(u),v\text{cos}(u),u>$ where $0\le u\le \frac{\pi }{2}$ and $0\le v\le 1$.

For a surface $z=g(x,y)$, we can define a parametric vector function using $\overrightarrow{r(x,y)}=<x,y,g(x,y)>$, hence we have $|\overrightarrow{r_x}\times \overrightarrow{r_y}|=\sqrt{1+g_x^2+g_y^2}$

$$\int {\int }_{S}f(x,y,z)dS=\int {\int }_{D}f(x,y,g(x,y))\sqrt{1+g_x^2+g_y^2}dA.$$

Example 3: Find $\int {\int }_{S}xdS$ where $S$ is the part of the plane $x+2y+z=6$ lies in the first octant.

Exercise 3: Find $\int {\int }_{S}ydS$ where $S$ is the part of the plane $2x+3y+z=12$ lies in the first octant.

Example 4: Find $\int {\int }_{S}xdS$ where $S$ is the surface $y=x^2+4z$ where $0\le x\le 1$ and $0\le z\le 1$.

Exercise 4: Find $\int {\int }_{S}zdS$ where $S$ is the surface $x=y+2z^2$ where $0\le y\le 1$ and $0\le z\le 1$.

Example 5: Find $\int {\int }_S(x-y-z)dS$ where $S$ is the part of half cylinder $y^2+z^2=4$ with $y\ge 0$ and lies between $x=-1$ and $x=1$.

Exercise 5: Find $\int {\int }_S(x+y-z)dS$ where $S$ is the part of half cylinder $x^2+z^2=4$ with $x\ge 0$ and lies between $y=0$ and $y=1$.

Example 6: Find $\int {\int }_S(xz)dS$ where $S$ is the boundary enclosed by the cylinder $y^2+z^2=4$ and the planes $x=0$ and $x=1$.

Group work:

1. Find $\int {\int }_S(yz)dS$ where $S$ is the boundary enclosed by the cylinder $x^2+y^2=4$ and the planes $z=0$ and $z=2$.

2. Find $\int {\int }_S(y)dS$ where $S$ is the triangular region bounded by three vertices: $(1,0,0)$, $(0,2,0)$ and $(0,0,3)$.

3. Find $\int {\int }_S{x}^{2}dS$ where $S$ has a vector equation $\overrightarrow{r(u,v)}=<u-2v,u+v+3,u++2v>$ where $0\le u\le 1$ and $0\le v\le 1$.