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Chapter 6 | Applications of Integration
12. y = e , y = e x , and y = e − x 13. y = | x | and y = x 2
31. y =4−3 x and y = 1 x 32. y = sin x , x =− π /6, x = π /6, and y =cos 3 x 33. y = x 2 −3 x +2and y = x 3 −2 x 2 − x +2 34. y =2cos 3 (3 x ), y =−1, x = π 4 , and x = − π 4 35. y + y 3 = x and2 y = x 36. y = 1− x 2 and y = x 2 −1 37. y =cos −1 x , y = sin −1 x , x =−1, and x =1 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
14. y = sin( πx ), y =2 x , and x >0 15. y =12− x , y = x , and y =1
16. y = sin x and y =cos x over x = [− π , π ] 17. y = x 3 and y = x 2 −2 x over x = [−1, 1] 18. y = x 2 +9and y =10+2 x over x = [−1, 3] 19. y = x 3 +3 x and y =4 x For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y -axis. 20. x = y 3 and x =3 y −2 21. x =2 y and x = y 3 − y 22. x =−3+ y 2 and x = y − y 2 23. y 2 = x and x = y +2 24. x = | y | and2 x =− y 2 +2 25. x = sin y , x =cos(2 y ), y = π /2, and y =− π /2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x -axis or y -axis, whichever
38. [T] x = e y and y = x −2 39. [T] y = x 2 and y = 1− x 2 40. [T] y =3 x 2 +8 x +9and3 y = x +24 41. [T] x = 4− y 2 and y 2 =1+ x 2 42. [T] x 2 = y 3 and x =3 y y = sin 3 x +2, y = tan x , x =−1.5, and x =1.5 44. [T] y = 1− x 2 and y 2 = x 2 43. 45. [T] y = 1− x 2 and y = x 2 +2 x +1 46. [T] x =4− y 2 and x =1+3 y + y 2 47. [T] y =cos x , y = e x , x =− π , and x =0
[T]
seems more convenient. 26. x = y 4 and x = y 5 27. y = xe x , y = e x , x =0, and x =1 28. y = x 6 and y = x 4 29. x = y 3 +2 y 2 +1and x =− y 2 +1 30. y = | x | and y = x 2 −1
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