Chapter 5: Double Integration
Section 5.7: Double Integration in Polar Coordinates
Example 5.7.8
Calculate the area that is common to the circle r=3 cosθ and the cardioid r=1+cosθ.
Solution
Mathematical Solution
Figure 5.7.8(a) shows the circle r=3 cosθ (in black), the cardioid r=1+cosθ (in red), and the region whose area is to be calculated, (in blue and green).
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.
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.
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:
Figure 5.7.8(a) Area to be found
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:
Figure 5.7.8(b) Animation: Sweep of green region
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:
Figure 5.7.8(c) Animation: Sweep of blue region
The area of the region shaded in blue and green in Figure 5.7.8(a) is
∫−π/3π/3∫01+ cosθr ⅆr ⅆθ+∫π/32 π/3∫03 cosθr dr dθ = 5 π4
Maple Solution - Interactive
Using the iterated double integral template in the Calculus palette, write
∫−π/3π/3∫01+cosθr ⅆr ⅆθ+∫π/32 π/3∫03 cosθr ⅆr ⅆθ = 5π4
evaluating the sum of the integrals via the Context Panel's option "Evaluate and Display Inline."
The appropriate angles for the bounds on θ can be gleaned from the calculations in Example 5.7.4.
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