Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
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Example 7.2.3
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Working in polar coordinates, calculate the area enclosed by the cardioid .
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Solution
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Mathematical Solution
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Figure 7.2.3(a) animates the polar area-element as it sweeps through the cardioid . The calculation of the area is based on the expression given in Table 7.2.1.
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>
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use plots, plottools in
module()
local R,X,Y,P,G;
P := plot(1+cos(theta), theta = 0 .. 2*Pi, coords = polar, scaling = constrained,color=black,tickmarks=[3,3]):
R := 1+cos(t):
X := unapply(R*cos(t), t):
Y := unapply(R*sin(t), t):
G:=proc(t)
display(polygon([[0,0],[X(t),Y(t)],[X(t+.1),Y(t+.1)]],color=red));
end:
print(animate(G,[theta],theta=0..2*Pi-.1,background=P,frames=62));
end module;
end use:
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Figure 7.2.3(a) Animation: area element for cardioid
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The calculation is simplified by noting that the integrals of and over the interval are both zero.
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Maple Solution
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A stepwise solution similar to the one at the left of Figure 7.2.3(a) can be obtained with the
tutor. Setting the Constant Multiple and Sum rules as Understood Rules helps eliminate some of the tedium of working through such a stepwise solution.
The area is most efficiently obtained interactively as follows.
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Expression palette: Definite Integral template
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Context Panel: Evaluate and Display Inline
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=
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