420 | | | | CHAPTER 6 APPLICATIONS OF INTEGRATION

,22. ,23. , , ,24. , ,25.

26.

27. , , ,28. , , ,

29–30 Use calculus to find the area of the triangle with the givenvertices.

, ,30. , ,

31–32 Evaluate the integral and interpret it as the area of aregion. Sketch the region.

31.

32.

33–34 Use the Midpoint Rule with to approximate thearea of the region bounded by the given curves.33. , ,34. , ,

; 35–38 Use a graph to find approximate -coordinates of the pointsof intersection of the given curves. Then find (approximately) thearea of the region bounded by the curves.35. ,36.

37. ,38. , y ­ x 10y ­ x cos x

y ­ x 32 3x 1 4y ­ 3x 2

2 2xy ­ e x, y ­ 2 2 x 2

y ­ x 4y ­ x sinsx 2d

x

x ­ 0y ­ xy ­ s3 16 2 x 3

0 ø x ø 1y ­ cos2spxy4dy ­ sin2spxy4d

n ­ 4

y40 |sx 1 2 2 x | dx

ypy20 | sin x 2 cos 2x | dx

s5, 1ds2, 22ds0, 5d

s21, 6ds2, 1ds0, 0d29.

x ù 04x 1 y ­ 4y ­ 8x 2y ­ 3x 2

x . 0y ­14 xy ­ xy ­ 1yx

y ­ | x |, y ­ x22 2

y ­ x 2, y ­ 2ysx 21 1d

0 ø x ø py ­ 1 2 cos xy ­ cos xx ­ py2x ­ 0y ­ sin 2xy ­ cos x

y ­ xy ­ sinspxy2d

x ­ y 22 1x ­ 1 2 y 2

21.1–4 Find the area of the shaded region.1. 2.

4.

5–28 Sketch the region enclosed by the given curves. Decidewhether to integrate with respect to x or y. Draw a typical approx-imating rectangle and label its height and width. Then find thearea of the region.5.

6.

7. ,8.

10.

11. ,12. ,

,14. , ,15. , ,16.

17. , ,18. , , ,19. ,20. , x ­ y4x 1 y2

­ 12x ­ 4 1 y 2x ­ 2y 2

x ­ 3x ­ 23y ­ x 2y ­ 8 2 x 2

x ­ 9y ­12 xy ­ sx

y ­ x 32 x, y ­ 3x

2py3 ø x ø py3y ­ 2 sin xy ­ tan x0 ø x ø 2py ­ 2 2 cos xy ­ cos x

y ­ x 22 6y ­ 12 2 x 2

13.

y ­ 4x 2 x 2y ­ x 2

y 2­ xy ­ x 2

y ­ 1 1 sx , y ­ s3 1 xdy3y ­ 1yx, y ­ 1yx 2, x ­ 29.

y ­ x 22 2x, y ­ x 1 4y ­ x 2y ­ x

y ­ sin x, y ­ e x, x ­ 0, x ­ py2y ­ x 1 1, y ­ 9 2 x 2, x ­ 21, x ­ 2

x

y

(_3,�3)

x=2y-¥

x=¥-4y

x

x=¥-2

x=e y

y=1

y=_1

y3.

x=2

y=œ„„„„x+2

y= 1x+1

x

y

y=x

y=5x-≈

x

y

(4,�4)

EXERCISES6.1