9.4 Areas and Lengths in Polar Coordinates

1-4. Find the area of the region that is bounded by the given curve and lies in the specified sector.


1. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-1.html 

 

              

theta=var('theta');

polar_plot(theta, (0, pi/2), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

  


 

r=theta;

A=integral(1/2*r^2,theta,0,pi/2);

1/48*pi^3


  2. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-2.html 

theta=var('theta');

polar_plot(e^(theta/3), (pi, 2*pi), fill=True).show(aspect_ratio=1, xmin=-5, xmax=10, ymin=-5, ymax=5) 

 


 

r=e^(theta/3);

A=integral(1/2*r^2,theta,pi,2*pi);

-3/4*e^(2/3*pi) + 3/4*e^(4/3*pi)


3. , .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-3.html 


theta=var('theta');

polar_plot(sqrt(theta), (0, pi/3), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

  

 

r=sqrt(theta);

A=integral(1/2*r^2,theta,0,pi/3);

1/36*pi^2


4. , .


5-9. Find the area bounded by one loop of the given curve.


5. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-5.html 

theta=var('theta');

polar_plot(2*cos(2*theta), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)


r=2*cos(2*theta);

A=integral(1/2*r^2,theta,pi/4,3*pi/4);

1/2*pi


6. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-6.html

theta=var('theta');

polar_plot(3*sin(3*theta), (0, pi), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

(3 leaves)

r=3*sin(3*theta);

A=integral(1/2*r^2,theta,0,pi/3);

3/4*pi


7. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-7.html 

theta=var('theta');

polar_plot(2*cos(4*theta), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

 

   (symmetric 8 leaves about -axis)


r=2*cos(4*theta);

A=integral(1/2*r^2,theta,pi/8,3*pi/8);

1/4*pi


8. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-8.html 

theta=var('theta');

polar_plot(2+sin(2*theta), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

  

  

r=2+sin(2*theta);

A=integral(1/2*r^2,theta,7*pi/6,11*pi/6);

 3/2*pi + 1/16*sqrt(3)


9. Find the area between a large loop and the enclosed small loop of the curve .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-9.html 

theta=var('theta');

polar_plot(1+2*cos(theta), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

   

 

r=1+2*cos(theta);

B=integral(1/2*r^2,theta,0,2*pi);

B

 3*pi


From #8, the area between a large loop and the enclosed small loop

of the curve is .


10-12. Sketch the curve and find the area that it encloses.


10. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-10.html        

              

theta=var('theta');

polar_plot(2*sin(3*theta), (0, pi), fill=True).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

 

  

 


r=2*sin(2*theta);

A=4*integral(1/2*r,theta,0,pi/2);

A

 3/2*pi


11. .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-11.html 

theta=var('theta');

polar_plot(3*(1-cos(2*theta)), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-8, xmax=8, ymin=-8, ymax=8)

   


r=3*(1-cos(2*theta));

A=integral(1/2*r^2,theta,0,2*pi);

A

27/2*pi


12. .


13-15. Find the area of the region that lies inside the first curve and outside the second curve.


13. , .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-13.html 

theta=var('theta');

p1=polar_plot(2*sin(theta), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

p2=polar_plot(1, (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

  

  

r=2*sin(theta)

A=2*integral(1/2*(r^2-1),theta,pi/6,pi/2);

A

1/3*pi + 1/2*sqrt(3)


14. , .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-14.html 

var('theta')

p1=polar_plot(3*sin(theta),(theta,0,2*pi), color="red", fill=True, fillcolor="red")

p2=polar_plot(1+cos(theta),(theta,0,2*pi), fill=True, fillcolor="blue")

show(p1+p2, aspect_ratio=1, ymin=-5, ymax=5, xmin=-5, xmax=5)

 

  

 


A=2*integral(1/2*((3*cos(theta))^2-(1+cos(theta))^2),theta,0,pi/3);

A

pi - 9/4*sqrt(3)


15. , .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-15.html 

theta=var('theta');

p1=polar_plot(2*cos(2*(theta)), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

p2=polar_plot(1,(0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

  

 


A=4*integral(1/2*(2*cos(2*theta)-1),theta,2*pi/6,4*pi/6);

A

-2/3*pi - 2*sqrt(3)


16-17. Find all points of intersection of the given curves and find the area of the region that lies inside both curves.


16. , .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-16.html 

theta=var('theta');

p1=polar_plot(sin(3*(theta)), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

p2=polar_plot(sin(theta),(0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

r=sin(3*theta);

A=2*integral(1/2*sin(theta)^2,theta,0,pi/4)+2*integral(1/2*r^2,theta,pi/4,pi/2);

A

1/4*pi - 1/3


17. , .

http://matrix.skku.ac.kr/cal-lab/cal-9-4-17.html 

theta=var('theta');

p1=polar_plot(cos(2*(theta)), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

p2=polar_plot(sin(2*(theta)),(0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

 

A=8*2*integral(1/2*sin(2*theta),theta,0,pi/8);

A

-2*sqrt(2) + 4


18-21. Find the length of the polar curve.


18. , .


19. , .


20.Graph the curve and find its length.

http://matrix.skku.ac.kr/cal-lab/cal-9-4-20.html 

              

theta=var('theta');

p1=polar_plot((cos(theta/3))^2, (0, 6*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)

 

 


r=var('r');

r=(cos(theta/3))^2;


drd(theta)=diff(r,theta);

drd(theta)

-2/3*sin(1/3*theta)*cos(1/3*theta)


L=4*integral(cos(theta/3)*sqrt(4/9*(sin(theta/3))^2+1),theta,0,3*pi/2);

L

2*sqrt(13) + 9*arcsinh(2/3)


21. Show that the area of the surface generated by rotating the polar curve with about the polar axis is .

 From a p365, we have the formula .

      Since ,

      .