Activity 13.12.2

In this activity, we will try to understand the scalar surface integral by looking at whether the value of the scalar surface integral will be positive, negative, or zero over common surfaces. In each part below, you are given a function and a surface. For each surface, first draw a plot of the surface and make sure you have labeled a proper scale for each coordinate direction. Then reason if the given surface integral is positive, negative, or zero. Be sure to justify your answers in terms of the function being integrated and the particulars of the surface of integration.

1. For $S_1$ defined as the top half ($z \geq 0$) of the sphere of radius one centered at the origin, consider the surface integral $\iint_{S_1} x \, dS$.

2. For $S_2$ defined as the bottom half ($z \leq 0$) of the sphere of radius one centered at the origin, consider the surface integral $\iint_{S_2} z \, dS$.

3. For $S_3$ defined as the disc of radius one centered at $(1,0,0)$ on the plane $x=1$, consider the surface integral $\iint_{S_3} x+z \, dS$.

4. For $S_1$ as defined above, consider the surface integral $\iint_{S_1} x+z \, dS$.