The region described in your problem is defined in cylindrical coordinates. I'll break down the problem based on the given information and convert everything into set notation using cylindrical coordinates.

Cylindrical Coordinates

Recall that the relationship between Cartesian and cylindrical coordinates is:

$x = r \cos(\theta), \quad y = r \sin(\theta), \quad z = z$

The problem provides some specific boundaries in the $xy$-plane and a surface bounding the region above, which I'll interpret and combine into set notation.

1. Top Boundary (Surface):

The top boundary is described by the surface:

$z = 6 - x^2 - y^2$

In cylindrical coordinates, this becomes:

$z = 6 - r^2$

2. Bottom Boundary (Lines in the $xy$-Plane):

The region in the $xy$-plane is bounded by the lines $y = x$, $y = -x$, and $y = -1$.

Let's convert these into cylindrical coordinates:

• $y = x$ becomes $\theta = \frac{3\pi}{4}$ (since $\tan^{-1}(1) = \frac{\pi}{4}$).
• $y = -x$ becomes $\theta = \frac{5\pi}{4}$ (since $\tan^{-1}(-1) = \frac{5\pi}{4}$).
• $y = -1$ becomes $r = -\frac{1}{\sin(\theta)}$ (since $y = r \sin(\theta) = -1$).

This defines a sector of the plane in the range of angles between $\frac{5\pi}{4} \leq \theta \leq \frac{3\pi}{4}$, and $r$ is bounded accordingly.

Set Notation in Cylindrical Coordinates

Now, we can express the region in set notation using cylindrical coordinates:

$F = \left\{ (r, \theta, z) \, \middle| \, 0 \leq r \leq -\frac{1}{\sin(\theta)}, \frac{5\pi}{4} \leq \theta \leq \frac{3\pi}{4}, 0 \leq z \leq 6 - r^2 \right\}$

Explanation:

1. Radial Bound: $0 \leq r \leq -\frac{1}{\sin(\theta)}$, determined by the line $y = -1$.
2. Angular Bound: $\frac{5\pi}{4} \leq \theta \leq \frac{3\pi}{4}$, from the lines $y = x$ and $y = -x$.
3. Vertical Bound: $0 \leq z \leq 6 - r^2$, defined by the surface on the top.

Would you like to clarify any part of this, or see how these boundaries affect the region graphically?

5 Related Questions:

1. How do cylindrical coordinates simplify the analysis of 3D regions?
2. How would you compute the volume of this region using triple integrals?
3. What are the advantages of using set notation in describing 3D regions?
4. Can you convert this region back into Cartesian coordinates explicitly?
5. How would you describe a solid bounded by $z = r^2$ instead of $z = 6 - r^2$?

Tip:

When using cylindrical coordinates, always check how the angular limits affect the radial and vertical boundaries, as these dependencies shape the region more than in Cartesian coordinates.