Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Graph the limaçon r=1/2−cosθ.
Figure 7.1.6(a) provides a slider-controlled drawing of the limaçon r=1/2−cosθ. As the slider advances, the graph of the limaçon is drawn dynamically in black, while the ray from the origin to the polar point r,θ is drawn in red.
The slider controls angle θ in degrees. As the slider moves to θ^, the limaçon is graphed for θ∈0,θ^, and the value of rθ^ is displayed.
The inner loop is graphed for r<0, the lower half when θ∈0 °,60 °; the upper, when θ∈300 °,360 °.
Figure 7.1.6(a) Dynamic drawing of the limaçon r=1/2−cosθ
A graph of the given limaçon as shown in Figure 7.1.6(b) can be obtained with the
In the first pane of the Interactive Plot Builder, make the selections shown in the upper portion of Figure 7.1.6(c). The Interactive Plot Builder defaults to an implicit plot.
Change the ranges for r and θ to those shown in Figure 7.1.6(c). Note that the range for r must include negative values because the right side of the equation r=1/2−cosθ actually becomes negative for θ∈0,π/3⋃5 π/3,2 π.
Then, click on the Options button, and change the coordinate system and the grid as per the lower portion of Figure 7.1.6(c).
The resulting graph will be Figure 7.1.6(b).
Figure 7.1.6(b) Graph of limaçon r=1/2−cosθ
Figure 7.1.6(c) Polar graph via the Plot Builder
The graph in Figure 7.1.6(b) is most easily drawn by invoking the Plot Builder (via the Context Panel) on 1/2−cosθ and selecting 2-D polar plot. The appropriate polar graph will then be drawn automatically.
(Unfortunately, re-executing this worksheet via the triple exclamation icon (!!!) in the toolbar will cause the following graph to be replaced with its Cartesian version. Should that happen, simply repeat the Plot Builder option of "2-D polar plot.")
Alternatively, execute either of the first two commands in Table 7.1.6(a) to obtain Figure 7.1.6(b). To obtain a graph of the limaçon on a rectangular Cartesian grid, execute the last command in the table. (Select Evaluate in the Context Panel.)
Table 7.1.6(a) Commands that will generate a graph of the limaçon r=1/2−cosθ
The conversion to Cartesian coordinates cannot be done naively because, unlike for the polar form of this limaçon, the right-hand side of
can become negative but the left-hand side cannot. An implicit Cartesian form of the limaçon is obtained if the square roots are eliminated by appropriate algebraic manipulations. For example, the equation
properly defines the limaçon implicitly in Cartesian coordinates.
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