Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
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Example 7.2.8
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Working in polar coordinates, calculate the area common to the circle and the inner loop of the limaçon .
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Solution
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Mathematical Solution
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Figure 7.2.8(a) shows the intersection of circle and the limaçon. The region whose area is to be calculated is divided into two subregions designated and .
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The (blue) dividing line connects the origin with the intersection point , whose Cartesian coordinates are .
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As per Figure 7.1.11(a), the limaçon is drawn with ; the circle, with .
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At , the limaçon starts at on the -axis, arrives at when , and at the origin when .
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use plots in
module()
local p1,p2,p3,p4,r1,r2;
r1:=sin(t);
r2:=1/5+cos(t);
p1:=implicitplot([r=r1,r=r2],r=-2..2,t=-Pi..Pi,coords=polar,gridrefine=3,scaling=constrained,color=[black,red]):
p2:=plot([[0,0],[12/25,9/25]],style=line, color=blue):
p3:=textplot({[.55,.4,typeset(P)],[.21,.24,typeset(R[1])],[.27,.14,typeset(R[2])]}):
p4:=plot([[12/25,9/25]],style=point,symbol=solidcircle,symbolsize=15,color=green):
print(display(p1,p2,p3,p4,scaling=constrained,labels=[x,y],tickmarks=[[0,.8,1.2],[-.6,0,1]]));
end module:
end use:
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Figure 7.2.8(a) Intersection of circle and limaçon
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The area of region is given by the integral
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On the circle, the arc bounding region is drawn with . Hence, the area of region is given by the integral
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The total area of the region common to the circle and the inner loop of the limaçon is then the sum .
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The astute reader will realize that the essence of this calculation is the determination of the bounding angles defining regions and , wherein the difficulty is the difference in these angles with respect to the inner loop of the limaçon, and the circle.
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Maple Solution
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Initialize
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Context Panel: Assign Name
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Context Panel: Assign Name
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Intersect the circle and limaçon
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Write the equations as shown, and press the Enter key.
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Context Panel: Solve≻Solve
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Point in Figure 7.2.8(a) therefore has polar coordinates . The other solution corresponds to the intersection of the outer loop of the limaçon and the circle.
The inner loop of the limaçon passes through the origin when , the solution of the equation in the interval . This is most easily found with the Roots command in the Student Calculus1 package:
Access from the Context Panel
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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Write the equation and press the Enter key.
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Context Panel: Student Calculus1≻Solve≻Find Roots
See Roots dialog in Figure 7.2.8(b)
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Figure 7.2.8(b) Roots dialog
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Access directly
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If the circle is drawn with , then it passes through point when , so .
Calculation of areas and and their sum
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