Problem 12 - Calculus Early Transcendentals

Question

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq \theta \leq \pi / 2, z=1\\}$$

Step-by-Step Solution

Answer

Given set: $$(r, \theta, z): 0\leq \theta \leq \pi / 2, z=1$$ Answer: The given set of points represents an infinitely extended quarter-circle (arc) lying on the z=1 plane in the first octant.

Step 1: Understand the given set

The given set of points in cylindrical coordinates is: $$(r, \theta, z): 0\leq \theta \leq \pi / 2, z=1$$For any (r, θ, z) in this set, the θ values range between 0 and the Π/2, and z equals 1. There is no constraint on r values, so r can be of any non-negative value (as by definition, r is non-negative in cylindrical coordinates).

Step 2: Sketch z=1 plane

Since the z-coordinate is constant and equal to 1, the shape will lie in the plane parallel to the xy-plane at a height of 1 unit. To draw this plane, make a rectangle parallel to the xy-plane and note that it will have a height of 1.

Step 3: Sketch 0 ≤ θ ≤ Π/2 region

Given that θ values range from 0 to Π/2, the points in this set will lie in the first octant (in Cartesian coordinates, the octant with all positive coordinates). This means that our shape will be limited to the region in the first octant of the z=1 plane. To sketch this region, starting from the origin, draw an arc with an angle of Π/2 on the z=1 plane.

Step 4: Sketch the shape without any constraint on r

Since there is no constraint on r values, r can take any non-negative value. This means that for a given angle θ in the specified range, r can extend to infinity. To convey this, draw curved arrows from the origin along the boundary of the region, indicating that the shape extends indefinitely along these lines in the first octant. The final sketch should represent a quarter-circle (arc) with curved arrows extending outward, all lying on the z=1 plane and in the first octant. This shape, in the cylindrical coordinate system, represents the given set of points.