Question 4.10: Describe the figure dened by (a) 1 ≤ r ≤ 2, θ = 60◦,−1 ≤ z ≤......
Describe the figure dened by
(a) 1 ≤ r ≤ 2, θ = 60◦,−1 ≤ z ≤ 1
(b) r = 1, 0◦ ≤ θ ≤ 90◦, 0 ≤ z ≤ 2
(a) Consider the r and θ coordinates first. The r coordinate varies from 1 to 2 while θ is fixed at 60◦. This represents the line PQ as shown in Figure 4.23. At P the value of r is 1; at Q the value of r is 2. The length of PQ is 1 and it is inclined at 60◦ to the x axis.
Now, we note that z varies from −1 to 1. We imagine the line PQ moving in the z direction from z = −1 to z = 1. This movement sweeps out a plane. Figure 4.24 illustrates this.
(b) The r coordinate is fixed at r = 1. The θ coordinate varies from 0◦ to 90◦. This produces the quarter circle, AB, as shown in Figure 4.25. At A, r = 1, θ = 0◦; at B, r = 1, θ = 90◦.
Examining the z coordinate, we see that z varies from 0 to 2. As z varies from 0 to 2, we imagine the curve AB sweeping out the curved surface as shown in Figure 4.26. At C, r = 1, θ = 0◦ , z = 2; at D, r = 1, θ = 90◦, z = 2. This surface is part of a cylinder.
If the range of values of a coordinate is not given it is understood that that variable varies across all its possible values. For example, a curve may be described by r = 1, z = −2. Here there is no mention of the values that θ can have. It is assumed that θ can have its full range of values, that is 0◦ to 360◦.
