Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Obtain the polar coordinates of the points of intersection of the curves defined by r=sinθ and r=1/5+cosθ, a circle and limaçon, respectively.
Figure 7.1.11(a) animates the tracing of the limaçon, with the circle fixed in each frame. The slider under the graph controls the value of θ in discrete jumps of 30 ° over the interval −π,π.
Table 7.1.11(a) lists, in the order of increasing θ, the four intersections r,θ of the two curves.
Table 7.1.11(a) Intersections of circle and limaçon
Figure 7.1.11(a) Circle and limaçon
The second and fourth intersections occur as the limaçon passes through the origin. For the circle, however, the origin is the point r,θ=0,0, whence there is no r,θ-pair for which the equations defining both curves are simultaneously satisfied.
As the slider in Figure 7.1.11(a) is moved, keep in mind that the polar angle starts from −π, and that for the top half of the inner loop of the limaçon, r is negative, so that π is added to each such negative polar angle prior to graphing the corresponding positive r.
Find the first and third intersections in Table 7.1.1(a)
Write the equations for the two given polar curves.
Context Panel: Solve≻Solve
Find θ for which the limaçon passes through the origin
Write the equation for r=0.
The solution Maple's solve command finds is approximately 101.54°; the second solution is its negative.
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