Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Graph the lemniscate r2=4 cos2 θ.
The graph of the lemniscate r2=4 cos2 θ appears in Figure 7.1.8(a). The left side of the defining equation is nonnegative; the right side is nonnegative for
θ∈0,π/4⋃3 π/4,5 π/4⋃7 π/4,2 π
If r∈0,2, then the lemniscate is traversed for
θ∈0,π/4⋃3 π/4,5 π/4⋃7 π/4,2 π
If r∈−2,2, then the lemniscate is traversed for
Figure 7.1.8(a) Lemniscate of Bernoulli
The convention for a negative radius mentioned at the end of the Introduction means that either branch r±= ±2cos2 θ contains the whole lemniscate for θ∈0,π/4⋃3 π/4,5 π/4⋃7 π/4,2 π.
Figure 7.1.8(b) is traced by a slider that controls the angle θ in the implicit polar representation of the lemniscate r2=4 cos2 θ.
The slider reads the angle in multiples of π/4 and reports the angle in degrees. Hence, θ takes on the discrete values k π/4,k=0,…,8, in the interval 0,2 π.
Table 7.1.8(a) lists the corresponding r,θ-pairs.
Table 7.1.8(a) r,θ-pairs for the lemniscate
Figure 7.1.8(b) Tracing the lemniscate
When cos2 θ is negative, r is actually imaginary.
A graph of the lemniscate 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
axis coordinates: polar
Smart graphing will automatically adjust ranges
r2=4 cos2 θ→
Table 7.1.8(b) lists a number of commands that will draw a graph of the lemniscate r2=4 cos2 θ. (Select Evaluate in the Context Panel.)
plots:-implicitplotr2=4 cos2 θ,r=0..2,θ=0..2 π,coords=polar,scaling=constrained
plots:-implicitplotr2=4 cos2 θ,r=−2..2,θ=0..π,coords=polar,scaling=constrained
plot2cos2 θ,θ=0..2 π,coords=polar,scaling=constrained
plot−2cos2 θ,θ=0..2 π,coords=polar,scaling=constrained
plot2cos2 θ,θ=0..2 π,coords=polar,scaling=constrained,axiscoordinates=polar
Table 7.1.8(b) Commands that will draw a graph of the lemniscate r2=4 cos2 θ
The fifth command in Table 7.1.8(b) draws the lemniscate on the coordinate curves for polar coordinates.
Finally, an implicitly drawn graph of the lemniscate can be obtained after a conversion to Cartesian coordinates based on the trig identity cos2 θ=2 cos2θ−1.
plots:-implicitplotx2+y2 = 8 x2x2+y2−4,x=−2..2,y=−1..1,gridrefine=3,scaling=constrained
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