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 limaçon can be drawn by applying the following to its equation of the. Clicking on the graph shown below will bring up the Plot Builder with all these options already selected. However, executing the complete worksheet with the !!! button in the toolbar will delete the graph drawn by the Plot Builder. If that's done, restore the original condition of the worksheet by closing the corrupted one and re-launching it.
Context Panel: Plots≻Plot Builder
2-D implicit plot
axis coordinates: polar
r →−1/2 to 3/2
θ →0 to 2*Pi
grid refine → 1
An alternative path to a graph of the given limaçon is via 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
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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