1. ESSENTIAL CALCULUS CH07 Applications of integration

2. In this Chapter: 7.1 Areas between Curves 7.2 Volumes 7.3 Volumes by Cylindrical Shells 7.4 Are Length 7.5 Applications to Physics and Engineering 7.6 Differential Equations Review

3. Chapter 7, 7.1, P359

4. Consider the region S that lies between two curves y=f(x) and y=g(x) and between the vertical lines x=a and x=b, where f and g are continuous functions and f(x)≥g(x) for all x in [a,b]. (See Figure 1.)

5. Chapter 7, 7.1, P359

7. Chapter 7, 7.1, P360

8. 2. The area A of the region bounded by the curves y=f(x), y=g(x), and the lines x=a, x=b, where f and g are continuous and f(x)≥g(x) for all x in [a,b], is

9. Chapter 7, 7.2, P365

10. Chapter 7, 7.2, P366 DEFINITION OF VOLUME Let S be a solid that lies between x=a and x=b. If the cross- sectional area of S in the plane P x, through x and perpendicular to the x-axis, is A(x), where A is an integrable function, then the volume of S is

11. Chapter 7, 7.2, P367

13. Chapter 7, 7.2, P368

15. Chapter 7, 7.2, P369

20. Chapter 7, 7.2, P370

22. Chapter 7, 7.2, P371

26. Chapter 7, 7.2, P372

29. Chapter 7, 7.3, P375

31. Chapter 7, 7.3, P376 2.The volume of the solid in Figure 3, obtained by rotating about the y-axis the region under the curve y=f(x) from a to b, is Where 0≤a<b

32. Chapter 7, 7.3, P377

34. Chapter 7, 7.3, P378

36. Chapter 7, 7.4, P380

39. If we let ∆y i =y i -y i-1, then

40. 2. THE ARC LENGTH FORMULA If f ’ is continuous on [a,b], then the length of the curve y=f(x), a≤x≤b, is Chapter 7, 7.4, P381

42. Chapter 7, 7.4, P382

43. Chapter 7, 7.4, P383

45. Chapter 7, 7.4, P384

47. Chapter 7, 7.5, P387 Work done in moving the object from a to b

48. Chapter 7, 7.6, P399 A differential equation is an equation that contains an unknown function and one or more of its derivatives. Here are some examples: 1. 2. 3.

49. Chapter 7, 7.6, P400 The order of a differential equation is the order of the highest derivative that occurs in the equation. A function f is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituted into the equation.

50. Chapter 7, 7.6, P400 A separable equation is a first-order differential equation that can be written in the form

51. Chapter 7, 7.6, P401 In many physical problems we need to find the particular solution that satisfies a condition of the form y(x 0 )=y 0. This is called an initial condition, and the problem of finding a solution of the differential equation that satisfies the initial condition is called an initial-value problem.

52. Chapter 7, 7.6, P403 where k is a constant. Equation 7 is called the logistic differential equation