1.9 Polar, Cylindrical, and Spherical Coordinates (continued)

Cylindrical Coordinates

Work through this exercise and explain your reasoning step by step.

Convert $(x,y,z)=(1,2,3)$ to cylindrical coordinates.

Use inequalities on cylindrical coordinates to describe the region given by the 6th octant.

Convert the cone $z^2=x^2+y^2$ to cylindrical coordinates. Write a couple of sentences to make sense of how you can simplify your conversion and describe the shape of the graph in terms of $z$ and $r$.

Draw a plot of the surface in $\R^3$ with equation $r=2$ in cylindrical coordinates. Write a couple of sentences about the shape and properties of this surface.

Draw a plot of the region given by $0 \leq \theta \leq \pi, 0 \leq r \leq 2, 0 \leq z \leq r^2$. Write a couple of sentences about the shape and properties of this region.

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 $0 \leq \theta \leq \pi$?

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

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

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?

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