1. Chapter 7-Applications of the Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

2. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Volumes by Slicing-The Method of Disks EXAMPLE: Calculate the volume V of a right circular cone which has height 11 and base of radius 5.

3. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solids of Revolution

4. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solids of Revolution THEOREM: (Method of Disks: Rotation about the x-axis) Suppose that f is a nonnegative, continuous function on the interval [a, b]. Let R denote the region of the xy-plane that is bounded above by the graph of f, below by the x- axis, on the left by the vertical line x = a, and on the right by the vertical line x = b. Then the volume V of the solid obtained by rotating R about the x-axis is given by

5. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solids of Revolution EXAMPLE: Calculate the volume of the solid of revolution that is generated by rotating about the x-axis the region of the xy-plane that is bounded by y = x 2, y = 0, x = 1, and x = 3.

6. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solids of Revolution EXAMPLE: Calculate the volume of the solid of revolution that is generated by rotating about the x-axis the region of the xy-plane that is bounded by y = x 2, y = 0, x = 1, and x = 3.

7. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solids of Revolution THEOREM: Suppose that g (y) is a nonnegative continuous function on the interval c ≤ y ≤ d. Let R denote the region of the xy-plane that is bounded by the graph of x = g(y), the y-axis, and the horizontal lines y = c and y = d. Then the volume V of the solid obtained by rotating R about the y-axis is given by

8. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solids of Revolution EXAMPLE: Calculate the volume enclosed when the graph of y = x 3, 2 ≤ x ≤ 4, is rotated about the y-axis.

9. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Method of Washers EXAMPLE: Let D be the region of the xy-plane that is bounded above by and below by y = x 2. Calculate the volume of the solid of revolution that is generated when D is rotated about the x-axis.

10. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Method of Washers THEOREM: (Method of Washers) Suppose that U and L are nonnegative, continuous functions on the interval [a, b] with L(x) ≤ U (x) for each x in this interval. Let R denote the region of the xy-plane that is bounded above by the graph of U, below by the graph of L, and on the sides by the vertical lines x = a and x = b. Then the volume V of the solid obtained by rotating R about the x-axis is given by

11. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Method of Washers EXAMPLE: Let R be the region of the xy-plane that is bounded above by y = e x, 0 ≤ x ≤1 and below by 0 ≤ x ≤ 1. Calculate the volume of the solid of revolution that is generated when R is rotated about the x-axis.

12. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Rotation about a Line that is Not a Coordinate Axis EXAMPLE: Rotate the parallelogram bounded by y = 3, y = 4, y = x, and y = x − 1 about the line x = 1 and find the resulting volume V.

13. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Steps in Calculating Volume by the Method of Slicing 1. Identify the shape of each slice. 2. Identify the independent variable which gives the position of each slice. 3. Write an expression, in terms of the independent variable, which describes the cross-sectional area of each slice. 4. Identify the interval [a, b] over which the independent variable ranges. 5. With respect to the independent variable of Step 2, integrate the expression for the cross-sectional area from Step 3 over the interval [a, b] from Step 4.

14. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Steps in Calculating Volume by the Method of Slicing EXAMPLE: Let V be the volume of a solid pyramid that has height h and rectangular base of area A. Then V = 1/3 Ah. Verify this formula for a solid pyramid of height 5 if the width and depth of the base are 2 and 3 respectively.

15. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Cylindrical Shells THEOREM: (The Method of Cylindrical shells: Rotation About the y-Axis) Let f be a nonnegative continuous function on an interval [a, b] of nonnegative numbers. Let V denote the volume of the solid generated when the region below the graph of f and above the interval [a, b] is rotated about the y-axis. Then

16. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Cylindrical Shells EXAMPLE: Calculate the volume generated when the region bounded by y = x 3 + x 2, the x- interval [0, 1], and the vertical line x = 1 is rotated about the y-axis.

17. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Cylindrical Shells THEOREM: (The Method of Cylindrical Shells: Rotation About the x-Axis) Suppose that 0 < c < d. Let g be a nonnegative continuous function on the interval [c, d] of nonnegative numbers. The volume of the solid generated when the region bounded by x = g (y), the y-axis, y = c, and y = d is rotated about the x-axis is

18. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Cylindrical Shells EXAMPLE: Let R be the region that is bounded above by the horizontal line y =  /2, below by the curve y =arcsin (x), 0 ≤ x ≤ 1, and on the left by the y-axis. Use the method of cylindrical shells to calculate the volume V of the solid that results from rotating R about the x-axis. EXAMPLE: Use the method of cylindrical shells to calculate the volume of the solid obtained when the region bounded by x = y 2 and x = y is rotated about the line y = 2.

19. Chapter 7-Application of the Integral 7.1 Volumes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. A region in the xy-plane is rotated about a vertical axis. If the method of disks is used to calculate the volume of the resulting solid of revolution, what is the variable of integration? 2. A region in the xy-plane is rotated about a horizontal axis. If the method of cylindrical shells is used to calculate the volume of the resulting solid of revolution, what is the variable of integration?

20. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Basic Method for Calculating Arc Length DEFINITION: If f has a continuous derivative on an interval containing [a, b], then the arc length L of the graph of f over the interval [a, b] is given by

21. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Examples of Arc Length EXAMPLE: Calculate the arc length L of the graph of f(x) = 2x 3/2 over the interval [0, 7]. EXAMPLE: Calculate the length L of the graph of the function f(x) = (e x + e −x )/2 over the interval [1, ln (8)].

22. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Examples of Arc Length DEFINITION: If g has a continuous derivative on an interval containing [c, d], then the arc length L of the graph of x = g(y) for c ≤ y ≤ d is given by EXAMPLE: Calculate the length L of that portion of the graph of the curve 9x 2 = 4y 3 between the points (0, 0) and (2/3, 1).

23. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parametric Curves DEFINITION: If  1 and  2 have continuous derivatives on an interval that contains I = [  ], then the arc length L of the parametric curve C = {(  1 (t),  2 (t)) :  ≤ t ≤  } is given by

24. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parametric Curves EXAMPLE: Calculate the length L of the parametric curve C defined by the parametric equations x = 1 + 2t 3/2 and y = 3 + 2t for 0 ≤ t ≤ 5.

25. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Surface Area DEFINITION: If f has a continuous derivative on an interval containing [a, b], then the surface area of the surface of revolution obtained when the graph of f over [a, b] is rotated about the x-axis is given by

26. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Surface Area EXAMPLE: Show that S = 4  r 2 is the surface area of a sphere of radius r.

27. Chapter 7-Application of the Integral 7.2 Arc Length and Surface Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. The arc length of the graph of y = x 3 between (0, 0) and (2, 8) is equal to for what function g (x)? 2. The arc length of the graph of the parametric curve x = e t, y = t 2 between (1, 0) and (e, 1) is equal to for what function g (t)? 3. The graph of y = x 3 between (0, 0) and (2, 8) is rotated about the x-axis. The area of the resulting surface of revolution is for what function g (x)? 4. What is the area of the surface of revolution that results from rotating the graph of y = mx, 0 ≤ x ≤ h about the x-axis?

28. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Basic Technique DEFINITION: Suppose that f is a Riemann integrable function on the interval [a, b]. The average value of f on the interval [a, b] is the number

29. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Basic Technique EXAMPLE: A rod of length 9 cm has temperature distribution T (x) =(2x − 6x 1/2 )  C for 0 ≤ x ≤ 9. This means that, at position x on the rod, the temperature is (2x − 6x 1/2 )  C Calculate the average temperature of the rod. EXAMPLE: What is the average value f avg of the function f(x) = x 2 on the interval [3, 6]?

30. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Random Variables DEFINITION: Suppose that X is a random variable all of whose values lie in an interval I. If there is a nonnegative function f such that for every subinterval [  ] of I, then we say that f is a probability density function of X. The abbreviation p.d.f. is commonly used. Sometimes the notation f X is used to emphasize the association between f and X.

31. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Random Variables EXAMPLE: In a large class, the grades on a particular exam are all between 38 and 98. Let X denote the score of a randomly selected student in the class. Suppose that the probability density function f for X is given by f (x) =(136x − 3724 − x 2 )/36000, 38 ≤ x ≤ 98. What is the probability that the grade on a randomly selected exam is between 72 and 82?

32. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Average Values in Probability Theory DEFINITION: If f is the probability density function of a random variable X that takes values in an interval I = [a, b], then the average (or mean) μ of X is defined to be This value is also said to be the expectation of X.

33. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Average Values in Probability Theory EXAMPLE: Let X denote the fraction of total impurities that are filtered out in a particular purification process. Suppose that X has probability density function f (x) = 20x 3 (1 − x) for 0 ≤ x ≤ 1. What is average of X?

34. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Population Density Functions EXAMPLE: In 1950 the population density of Tulsa was given by f (x) = 28000 e −4x/5, where x represents the distance in miles from the central business district. About how many people lived within 20 miles of the city center?

35. Chapter 7-Application of the Integral 7.3 The Average Value of a Function Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What is the average of sin (x) over the interval [0,  ]? 2. For what c in I = [0, 3] is f (c) the average value of f (x) = x 2 on I? 3. Suppose that the probability density of a nonnegative random variable X is f (x) = exp (−x), 0 ≤ x < . What is the probability that X ≤ 1? 4. What is the mean of a random variable that has probability density function f (x) = x/2 for 0 ≤ x ≤ 2?

36. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Moments DEFINITION: Let c be any real number. Suppose that f is continuous and nonnegative on the interval [a, b]. Let R denote the planar region bounded above by the graph of y = f (x), below by the x-axis, and on the sides by the line segments x = a and x = b. If R has a uniform mass density , then the moment M x=c of R about the axis x = c is defined by

37. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Moments EXAMPLE: Let R be the region bounded by y = x − 1, y = 0, and x = 6. Suppose that R has uniform mass density  = 2. Calculate the moments about the axes x = 5 and x = 0.

38. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Center of Mass DEFINITION: Let R be a region as shown below. Then the center of mass of R is the point (x, y) whose coordinates satisfy

39. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Center of Mass THEOREM: Let f be a continuous nonnegative function on the interval [a, b]. Let R denote the region bounded above by the graph of y = f (x), below by the x-axis, and on the sides by the line segments x = a and x = b. Let M denote the mass of R. If R has a uniform mass density , then the x-coordinate of the center of mass of R is given by and the y-coordinate of the center of mass of R is given by

40. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Center of Mass EXAMPLE: Let R be the region bounded by the lines y = x−1, y = 0, and x = 6. Suppose that R has uniform mass density  = 2. Calculate the center of mass of R.

41. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Center of Mass THEOREM: Let f and g be continuous functions on the interval [a, b] with g (x) ≤ f (x) for all x in [a, b]. Let R denote the region bounded above by the graph of y = f (x), below by the graph of y = g (x), and on the sides by the line segments x = a and x = b. If R has uniform mass density, then the x-coordinate of the center of mass of R is given by The y-coordinate of the center of mass of R is given by

42. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Center of Mass EXAMPLE: Let R be the region bounded above by y = x + 1 and below by y = (x − 1) 2. Suppose that R has unit mass density. What is the center of mass of R?

43. Chapter 7-Application of the Integral 7.4 Center of Mass Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1 Let R denote the triangle with vertices (0, 0), (2, 0), and (2, 6). The x-center of mass of R is given by the equation for what function g (x)? 2. Let R denote the triangle with vertices (0, 0), (2, 0), and (2, 6). The x-center of mass y of R is given by the equation for what function g (x)?

44. Chapter 7-Application of the Integral 7.5 Work Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using Integrals to Calculate Work DEFINITION: Suppose that a body is moved linearly from x = a to x = b by a force in the direction of motion. If the magnitude of the force at each point x in [a, b] is F(x), then the total work W performed is

45. Chapter 7-Application of the Integral 7.5 Work Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Examples with Weights that Vary EXAMPLE: A man carries a leaky 50 pound sack of sand straight up a 100 foot ladder that runs up the side of a building. He climbs at a constant rate of 20 feet per minute. Sand leaks out of the sack at a rate of 4 pounds per minute. Ignoring the man’s own weight, how much work does he perform on this trip?

46. Chapter 7-Application of the Integral 7.5 Work Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Example Involving a Spring EXAMPLE: If 5 J work is done in extending a spring 0.2 m beyond its equilibrium position, then how much additional work is required to extend it a further 0.2 m?

47. Chapter 7-Application of the Integral 7.5 Work Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Examples that Involve Pumping a Fluid from a Reservoir EXAMPLE: A tank full of water is in the shape of a hemisphere of radius 20 feet. A pump floats on the surface of the water and pumps the water from the surface to the top of the tank, where the water just runs off. How much work is done in emptying the tank?

48. Chapter 7-Application of the Integral 7.5 Work Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. To overcome friction, a force of 12 − 3x 2 N is used to push an object from x = 0 to x = 2 m. How much work is done? 2. If the amount of work in stretching a spring 0.02 meter beyond its equilibrium position is 8 J, then what force is necessary to maintain the spring at that position? 3. A cube of side length 1 m is filled with a fluid that weighs 1 newton per cubic meter. What work is done in pumping the fluid to the surface?

49. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solutions of Differential Equations DEFINITION: We say that a differentiable function y is a solution of the differential equation dy/dx=F(x,y) if y’(x) = F (x, y (x)) for every x in some open interval. The graph of a solution is said to be a solution curve of the differential equation.

50. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solutions of Differential Equations EXAMPLE: Let C denote an arbitrary constant. Verify that the function y (x) = x+Ce −x −1 is a solution of the differential equation dy/dx= x − y.

51. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Slope Fields EXAMPLE: Sketch a slope field for the differential equation, dy/dx= x − y.

52. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Initial Value Problem DEFINITION: If x 0 and y 0 are specified values, then the pair of equations is said to be an initial value problem (often abbreviated to “IVP”). We say that a differentiable function y is a solution of the initial value problem above if y(x 0 ) = y 0 and y’(x) = F (x, y (x)) for all x in some open interval containing x 0.

53. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Initial Value Problem EXAMPLE: Solve the initial value problem dy/dx = x − y, y (0) = 2.

54. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Separable Equations EXAMPLE: Solve the initial value problem

55. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Equations of the form dy/dx=g(x) THEOREM: If g is a continuous function on an open interval containing a then the initial value problem has a unique solution. It is

56. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Examples from the Physical Sciences EXAMPLE: According to Torricelli’s Law, the rate at which water flows out of a tank from a small hole in the bottom is proportional to the area A of the hole and to the square root of the height y (t) of the water in the tank. That is, there is a positive constant k such that where V (t) is the volume of water in the tank at time t. Consider a cylindrical tank of height 2.5 m and radius 0.4 m that has a hole 2 cm in diameter on its bottom. Suppose that the tank is full at time t = 0. If k = 2.6 m 1/2 /s, find the height y (t) of water as a function of time. How long will it take for the tank to empty?

57. Chapter 7-Application of the Integral 7.6 First Order Differential Equations-Separable Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. To which of the following differential equations can we apply the method of separation of variables: A) dy/dx =exp (xy) B) dy/dx = exp (x + y) C) dy/dx = ln (x · y) D) dy/dx = ln (x + y) ? 2. Solve 3. Solve the initial value problem dy/dx = x/y, y (2) = 3. 4. If dy/dx = sin(x 3 ) and y(  )=2, then for what , and C is

58. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solving Linear Differential Equations THEOREM: Suppose that p (x) and q (x) are continuous functions. Let P (x) be any antiderivative of p (x). Then the general solution of the linear equation is where C is an arbitrary constant.

59. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Solving Linear Differential Equations EXAMPLE: Solve the initial value problem EXAMPLE: Find the general solution of the linear differential equation

60. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application: Mixing Problems EXAMPLE: A 200 gallon tank is filled with a salt solution initially containing 10 pounds of salt. An inlet pipe brings a solution of salt in at the rate of 10 gallons per minute. The concentration of salt in the incoming solution is 1−e −t/60 pounds per gallon when t is measured in minutes. An outlet pipe prevents overflow by allowing 10 gallons per minute to flow out of the tank. How many pounds of salt are in the tank at time t? Long term, about how many pounds of salt will be in the tank?

61. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Linear Equations with Constant Coefficients THEOREM: Suppose a and b are constants with b 0. Then the linear equation has general solution The initial value problem Has unique solution

62. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Newton’s Law for Temperature Change THEOREM: Suppose that T 0 = T (0) is the temperature of an object when it is placed in an environment that has constant temperature T . Suppose that the temperature T (t) of the object is governed by Newton’s law of temperature change. That is, suppose that T (t) is a solution of equation then

63. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Newton’s Law for Temperature Change EXAMPLE: A thermometer is at room temperature (20.0  C). One minute after being placed in a patient’s throat it reads 38.0  C. One minute later it reads 38.3  C. Is this second reading an accurate measure (to three significant digits) of the patient’s temperature?

64. Chapter 7-Application of the Integral 7.7 First Order Differential Equations-Linear Equations Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What is the integrating factor for the linear differential equation 2. If P (x) =  p (x) dx, then what is the antiderivative of 3. What is the general solution of 4. If y = 2 + Ce −3t is the general solution of then what are  and  ?