Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Example 7.1.7
Graph the limaçon r=3/2−cosθ.
Solution
A graph of the limaçon can be drawn by applying the following to its equation. 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
2-D Options, then immediately back to Basic Options coordinates: polar axis coordinates: polar Smart graphing will automatically adjust ranges
r=3/2−cosθ→
An alternate 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 7.1.7(b). Because the Interactive Plot Builder was launched on an equation, it defaults to an implicit plot.
Change the ranges for r and θ to those shown in Figure 7.1.7(b).
Then, click on the Options button, and change the coordinate system and the grid as per the lower portion of Figure 7.1.7(b).
The resulting graph will be Figure 7.1.7(a).
Figure 7.1.7(a) Limaçon r=3/2−cosθ
Figure 7.1.7(b) Polar graph via the Plot Builder
Alternatively, execute either of the first two commands in Table 7.1.7(a) to obtain Figure 7.1.7(a). To obtain a graph of the limaçon on a rectangular Cartesian grid, execute either of the last two commands in the table. (Select Evaluate in the Context Panel.)
plot3/2−cosθ,θ=0..2 π,coords=polar,axiscoordinates=polar,scaling=constrained
plots:-polarplot3/2−cosθ,θ=0..2 π,scaling=constrained
plots:-implicitplotr=3/2−cosθ,r=1/2..5/2,θ=0..2 π,coords=polar,gridrefine=3
plot3/2−cosθ,θ=0..2 π,coords=polar,scaling=constrained
Table 7.1.7(a) Commands that will generate a graph of the limaçon r=3/2−cosθ
Because the right-hand side remains positive, the conversion to Cartesian coordinates is straightforward. The following command will graph the implicit Cartesian form of this limaçon.
plots:-implicitplotx2+y2=3/2−x/x2+y2,x=−3..1,y=−2..2,gridrefine=3,scaling=constrained
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