91Ó°ÊÓ

Cylindrical coordinates \((r, \theta, z)\) are a way to describe points in 3D space. They are related to Cartesian coordinates \((x, y, z)\) as follows: $$x = r \cos \theta$$ $$y = r \sin \theta$$ $$z = z$$ In cylindrical coordinates, \(r \geq 0\) represents the distance from the \(z\)-axis, \(\theta\) is the angle between the projection of the radial vector on the \(xy\)-plane and the positive \(x\)-axis, and \(z\) is the height along the \(z\)-axis.

In the given set, we have three parameters: \(r\), \(\theta\), and \(z\). The constraints for these parameters are: - \(0 \leq \theta \leq \pi / 2\): The angle \(\theta\) ranges from \(0\) to \(\pi / 2\). - \(z = 1\): The height \(z\) is a constant value of \(1\). There is no constraint given for the parameter \(r\). Therefore, it can take any value greater than or equal to zero.

To sketch the set, follow these guidelines: 1. Since \(\theta\) ranges from \(0\) to \(\pi / 2\), the radial lines will be confined to the first quadrant in the \(xy\)-plane. 2. The constant height \(z=1\) means that the formed shape will be parallel to the \(xy\)-plane and located one unit above it. 3. Since there is no constraint on \(r\), radial lines will extend to infinity. With these guidelines, we can sketch the set as an infinite quarter plane in the first quadrant, parallel to the \(xy\)-plane, and one unit above it. -A summed up action plan for sketching the set is as follows: 1. Draw the \(x\), \(y\), and \(z\) axes. 2. Draw the radial lines with the angle \(\theta\) ranging from \(0\) to \(\pi / 2\) and extending to infinity. 3. Locate the set one unit above the \(xy\)-plane and parallel to it. 4. Shade the resulting infinite quarter plane.