Activity 9.9.9

In this activity, we graph some surfaces using cylindrical coordinates. To improve your intuition and test your understanding, you should first think about what each graph should look like before you plot it using appropriate technology.

1. What familiar surface is described by the points in cylindrical coordinates with $r=2$, $0 \leq \theta \leq 2\pi$, and $0 \leq z \leq 2$? How does this example suggest that we call these coordinates cylindrical coordinates? How does the surface change if we restrict $\theta$ to $0 \leq \theta \leq \pi$?

2. What familiar surface is described by the points in cylindrical coordinates with $\theta=2$, $0 \leq r \leq 2$, and $0 \leq z \leq 2$?

3. What familiar surface is described by the points in cylindrical coordinates with $z=2$, $0 \leq \theta \leq 2\pi$, and $0 \leq r \leq 2$?

4. Plot the graph of the cylindrical equation $z=r$, where $0 \leq \theta \leq 2\pi$ and $0 \leq r \leq 2$. What familiar surface results?

5. Plot the graph of the cylindrical equation $z= \theta$ for $0 \leq \theta \leq 4 \pi$. What does this surface look like?