10.

4 A R E A S A N D L E N G T H S I N P O L A R C O O R D I N AT E S

1– 8 Find the area of the region that is bounded by the given 21. r 苷 sin 5␪ 22. r 苷 2 ⫹ 3 cos ␪ (inner loop)
curve and lies in the specified sector.
1. r 苷 ␪, 0艋␪艋␲ 23–24 Find the area of the region that lies inside the first curve and
␪
2. r 苷 e , ⫺␲兾2 艋 ␪ 艋 ␲兾2 outside the second curve.
23. r 苷 1 ⫺ cos ␪, r 苷 32
3. r 苷 2 cos ␪, 0 艋 ␪ 艋 ␲兾6
24. r 苷 3 cos ␪, r 苷 2 ⫺ cos ␪
4. r 苷 1兾␪, ␲兾6 艋 ␪ 艋 5␲兾6
5. r 苷 sin 2␪, 0 艋 ␪ 艋 ␲兾6
25. Find the area inside the larger loop and outside the smaller loop
6. r 苷 cos 3␪, ⫺␲兾12 艋 ␪ 艋 ␲兾12 of the limaçon r 苷 3 ⫹ 4 sin ␪.
7. r 苷 3 sin ␪, ␲兾4 艋 ␪ 艋 3␲兾4 ; 26. Graph the hippopede r 苷 s1 ⫺ 0.8 sin 2␪ and the circle
r 苷 sin ␪ and find the exact area of the region that lies inside
8. r 苷 ␪ 2, ␲兾2 艋 ␪ 艋 3␲兾2 both curves.

27–32 Find the length of the polar curve.

9–16 Sketch the curve and find the area that it encloses.
27. r 苷 5 cos ␪, 0 艋 ␪ 艋 3␲兾4
9. r 苷 5 sin ␪ 10. r 苷 4 ⫺ sin ␪
28. r 苷 2␪, 0 艋 ␪ 艋 2␲
11. r 苷 sin 3␪ 12. r 苷 4共1 ⫺ cos ␪ 兲
29. r 苷 1 ⫹ cos ␪
13. r 苷 2 cos ␪ 14. r 苷 1 ⫹ sin ␪
30. r 苷 e ⫺␪, 0 艋 ␪ 艋 3␲
15. r 苷 3 ⫺ cos ␪ 16. r 苷 sin 4␪
31. r 苷 cos 共␪兾4兲
2

32. r 苷 cos 2共␪兾2兲

; 17. Graph the curve r 苷 2 ⫹ cos 6␪ and find the area that it
encloses.
; 18. The curve with polar equation r 苷 2 sin ␪ cos ␪ is called a
2
33–34 Use a calculator or computer to find the length of the loop
bifolium. Graph it and find the area that it encloses. correct to four decimal places.

19 –22 Find the area of the region enclosed by one loop of 33. One loop of the four-leaved rose r 苷 cos 2␪
the curve. 34. The loop of the conchoid r 苷 4 ⫹ 2 sec ␪
19. r 苷 cos 3␪ 20. r 苷 3 sin 2␪
 10.4 ANSWERS

3
 π −π

1. 16 π 2. 14 e − e 17.

√
3. π
6
+ 43 12
4. 5π

√
5. 4π−3
96
3 1
6. 24 (π + 2)

9π
7. 98 (π + 2) 8. 121
160
π5 2

18.

9. 10.

π
8
π
19. 12 20. 9π
8
  √
25
π 33π
π
21. 20 22. 17
2
cos−1 32 − 3 5
4 2
√ √
23. 9 8 3 − 14 π 24. 3 3
−1 3
  √
11. 12. 25. 34 sin 4
+9 7
26. 1.1

⫺1.1 1.1

24π
⫺0.6
π
3 1
√ √
4
10
π − 10
arcsin 35 − 1
5
5

  
13. 14.
15
1 + ln2 2 22π − 1
27. 4
π 28.
ln 2
√  −3π

(2, 0) 29. 8 30. 2 1 − e
O
31. 16
3
32. 4
O
33. 2.4221 34. 5.8128

π 3π
2

15. 共3, ␲ 兾2兲

16.

(4, ␲兲
O 共2, 0兲

π
19π
2
2