Calculus-Sec-10-4-Solution
10.4 Areas and Lengths in Polar Coordinates by SGLee - HSKim - JHLee
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
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
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
.
From figure, the area between a large loop and the enclosed small loop
of the curve is .
10. .
http://matrix.skku.ac.kr/cal-lab/cal-9-4-10.html
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
.
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
18-21. Find the length of the polar curve.
18. ,
19. ,
21. Show that the area of the surface generated by rotating the polar curve
with about the polar axis is .
From http://mathworld.wolfram.com/SurfaceofRevolution.html, we have the formula
.
Since ,
.