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Chapter 5: Double Integration
Section 5.7: Double Integration in Polar Coordinates
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Example 5.7.8
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Calculate the area that is common to the circle and the cardioid .
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Solution
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Mathematical Solution
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Figure 5.7.8(a) shows the circle (in black), the cardioid (in red), and the region whose area is to be calculated, (in blue and green).
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Figure 5.7.8(b) is an animation in which the "green region" is swept by the polar ray from the origin to the cardioid.
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Figure 5.7.8(c) is an animation is which the "blue region" is swept by the polar ray from the origin to the circle.
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use plots in
module()
local p1,p2,p3,p4,p5,p6,p7,R,R1,R2;
R:=sqrt(x^2+y^2);
R1:=3*cos(t);
R2:=1+cos(t);
p1:=plot([R1,R2],t=0..2*Pi,coords=polar,color=[black,red],thickness=[1,3]);
p2:=inequal([R<=1+x/R],x=3/4..3,y=-1.5..1.5,color=green);
p3:=shadebetween(sqrt(3)*x,-sqrt(3)*x,x=0..3/4,color=green,transparency=0):
p4:=plot([sqrt(3)*x,-sqrt(3)*x],x=0..3/4);
p5:=shadebetween(sqrt(3*x-x^2),sqrt(3)*x,x=0..3/4,color=blue);
p6:=shadebetween(-sqrt(3)*x,-sqrt(3*x-x^2),x=0..3/4,color=blue);
p7:=display(p1,p2,p3,p4,p5,p6,scaling=constrained,labels=[x,y],tickmarks=[4,3],size=[400,400]);
print(p7);
end module:
end use:
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Figure 5.7.8(a) Area to be found
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use plots in
module()
local q1,q2,q3,q4,q5,a,b,g,G;
a:=-Pi/3;
b:=Pi/3;
g:=t->0;
G:=t->1+cos(t);
q1 := plot([[g(t),t,t=a..b],[G(t),t,t=a..b],[r,a,r=g(a)..G(a)],[r,b,r=g(b)..G(b)]],coords=polar, color=[red,green,blue,blue]):
q2 := animate(plot, [[k*g(theta)+(1-k)*G(theta),theta,k=0..1],coords=polar,color="Orange",thickness=3],theta=a..b,frames=25):
q3 := animate(plot,[[[g(theta),theta]],style=point,coords=polar, symbol=circle,color=black,symbolsize=20],theta=a..b,frames=25):
q4 := animate(plot,[[[G(theta),theta]],style=point,coords=polar, symbol=solidcircle,color=black,symbolsize=20],theta=a..b,frames=25):
q5:=display([q1,q2,q3,q4],scaling=constrained,labels=[x,y]);
print(q5);
end module:
end use:
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Figure 5.7.8(b) Animation: Sweep of green region
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use plots in
module()
local q1,q2,q3,q4,q5,a,b,g,G;
a:=Pi/3;
b:=2*Pi/3;
g:=t->0;
G:=t->3*cos(t);
q1 := plot([[g(t),t,t=a..b],[G(t),t,t=a..b],[r,a,r=g(a)..G(a)],[r,b,r=g(b)..G(b)]],coords=polar, color=[red,green,blue,blue]):
q2 := animate(plot, [[k*g(theta)+(1-k)*G(theta),theta,k=0..1],coords=polar,color="Orange",thickness=3],theta=a..b,frames=25):
q3 := animate(plot,[[[g(theta),theta]],style=point,coords=polar, symbol=circle,color=black,symbolsize=20],theta=a..b,frames=25):
q4 := animate(plot,[[[G(theta),theta]],style=point,coords=polar, symbol=solidcircle,color=black,symbolsize=20],theta=a..b,frames=25):
q5:=display([q1,q2,q3,q4],scaling=constrained,labels=[x,y]);
print(q5);
end module:
end use:
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Figure 5.7.8(c) Animation: Sweep of blue region
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The area of the region shaded in blue and green in Figure 5.7.8(a) is
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=
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Maple Solution - Interactive
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Using the iterated double integral template in the Calculus palette, write
=
evaluating the sum of the integrals via the Context Panel's option "Evaluate and Display Inline."
The appropriate angles for the bounds on can be gleaned from the calculations in Example 5.7.4.
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