Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Example 7.1.5
Graph the cardioid r=21−cosθ.
Solution
Mathematical Solution
Given that the cardioid has a characteristic heart-like shape, it is enough to know its orientation, its intercepts, and that it passes through the origin. The two y-intercepts occur at θ=π/2 and θ=3 π/2; the x-intercept, at θ=π. Consequently, rπ/2=2, rπ=4, and r3 π/2=2. This would be enough data for sketching a graph of the cardioid. Figure 7.1.5(a) is a Maple-drawn graph of the cardioid, graphed as a polar curve.
Alternatively, convert to Cartesian coordinates, obtaining
x2+y2=21−xx2+y2
Figure 7.1.5(a) Cardioid r=21−cosθ
which can be graphed with any tool that handles implicitly defined functions.
Maple Solution
Figure 7.1.5(a) can be drawn by applying the following to the equation of the cardioid. 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=21−cosθ→
Alternatively, use the .
In the first pane of the Interactive Plot Builder, make the selections shown in the upper portion of Figure 7.1.5(b). The Interactive Plot Builder defaults to an implicit plot.
Change the ranges for r and θ to those shown in Figure 7.1.5(b).
Then, click on the Options button, and in the lower-left corner change the coordinate system to the option shown in the lower portion of Figure 7.1.5(b).
The resulting graph will be Figure 7.1.5(a).
Figure 7.1.5(b) Polar graph via the Plot Builder
Alternatively, execute either of the first two commands in Table 7.1.5(a) to obtain Figure 7.1.5(a). To obtain a graph of the cardioid on a rectangular Cartesian grid, execute either of the second two commands in the table. (Select Evaluate in the Context Panel.)
plot21−cosθ,θ=0..2 π,coords=polar,axiscoordinates=polar,scaling=constrained
plots:-polarplot21−cosθ,θ=0..2 π,scaling=constrained
plots:-implicitplotr=21−cosθ,r=0..4,θ=0..2 π,coords=polar,gridrefine=3,scaling=constrained
plot21−cosθ,θ=0..2 π,coords=polar,scaling=constrained
Table 7.1.5(a) Commands that will generate a graph of the cardioid r=21−cosθ
To graph the Cartesian equivalent of the cardioid, use the following command. (Select Evaluate in the Context Panel.)
plots:-implicitplotx2+y2=21−xx2+y2,x=−4..1,y=−3..3,gridrefine=3, scaling=constrained
Context Panel: Plots≻2-D Implicit Plot≻x,y applied to the Cartesian equation will also graph the cardioid.
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