Chapter 5: Double Integration
Section 5.7: Double Integration in Polar Coordinates
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Example 5.7.6
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Calculate the area that is inside the large loop, but outside the small inner loop, of the limaçon .
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Solution
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Mathematical Solution
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Figure 5.7.6(a) shows the limaçon, and, in green, the region whose area is to be calculated. The area of the inner loop, shaded in red, is to be excluded. This is a difficult calculation because of the way the graph is drawn with respect to the angle . The approach here will be to find the total area inside the region, that is, the sum of what is shaded in red and green, then to subtract the area that is shaded red. The total area can be found by doubling the area that lies above the horizontal axis, .
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Figure 5.7.6(b) is an animation in which the limaçon is drawn under the action of the slider in the animation toolbar. The polar angle appears above the vertical axis. Use this animation to infer the appropriate angles for the iterated double integral that follows.
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Inspired by the animation in Figure 5.7.6(b), Figure 5.7.6(c) is a graph of . It suggests that the inner loop is drawn for , that is, for the interval between the two zeros of .
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use plots in
module()
local p1,p2,p3,R,L;
R:=sqrt(x^2+y^2);
L:=1/2+cos(t);
p1:=inequal(R<=1/2+x/R,x=-.2..3/2,y=-1..1,scaling=constrained,color=green,transparency=.5,thickness=3);
p2:=plot([L*cos(t),L*sin(t),t=2*Pi*(1/3)..4*Pi*(1/3)],filled=[color=red],color=black,thickness=3);
p3:=display(p2,p1,tickmarks=[3,3],scaling=constrained,labels=[x,y],tickmarks=[3,3]);
print(p3);
end module:
end use:
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Figure 5.7.6(a) Region
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Figure 5.7.6(b) Animation
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Figure 5.7.6(c) Zeros
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The area of the region shaded in green in Figure 5.7.6(a) is
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= ≐ 2.08
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Maple Solution - Interactive
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The task template in Tables 5.7.6(a) and 5.7.6(b) can be used to visualize the region of integration over which a given iterated integral acts. Table 5.7.6(a) calculates the total area in the upper half of the limaçon; the second, the area of the complete inner loop. In each case, select an order of integration, and provide an integrand of 1 to compute area. Supply the limits of integration, and use the Exact button to obtain the value of the iterated integral, and the Draw Graphs button to obtain the two figures provided by the task template.
Assign the limaçon a name
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Context Panel: Assign Name
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Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Polar
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Evaluate and Graph
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Area Element
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Value of Integral
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Table 5.7.6(a) Task template for visualizing integration over upper half of the limaçon
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The figure on the left is an animation that shows how the radial cone representing traverses the region of integration. The figure on the right is a representation of the volume of a solid of height 1, with base the region of integration. Since the height is 1, the number computed for the volume is the same number as the area. If this figure is rotated and viewed from above, it appears to be a shaded version of the region of integration. These visual clues help to decide of the polar area has been properly identified and calculated.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Polar
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Evaluate and Graph
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Area Element
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Value of Integral
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Table 5.7.6(b) Task template for visualizing integration over inner loop of the limaçon
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Combining the results in Tables 5.6.7(a - b) gives the requisite area as
= ≐ 2.08
The details of an interactive calculation of the required area appear in Table 5.7.6(c).
Initialize
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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Find the zeros of
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Context Panel:
Student Calculus1≻Solve≻Find Roots
Complete the dialog as per Figure 5.7.6(d)
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Figure 5.7.6(d) Roots dialog
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Implement and evaluate the iterated integration
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Calculus palette: Iterated double integral template
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Context Panel: Evaluate and Display Inline
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=
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Table 5.7.6(c) Details of the interactive calculation of the required area.
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