chapter 7-applications of the integral calculus, 2ed, by blank & krantz, copyright 2011 by john...

Chapter 7-Applications of the Integral

Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 7-Application of the Integral7.1 Volumes


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.

Chapter 7-Application of the Integral

7.1 Volumes


Solids of Revolution


7.1 Volumes



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


7.1 Volumes



EXAMPLE: Calculate the volume of the solid of revolution that is generated by rotating about the x-axis the regionof the xy-plane that is bounded by y = x2, y = 0, x = 1, and x = 3.


7.1 Volumes



THEOREM: Suppose that g (y) is a nonnegative continuousfunction 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 aboutthe y-axis is given by


7.1 Volumes



EXAMPLE: Calculate the volume enclosed when the graph of y = x3, 2 ≤ x ≤ 4, is rotated about the y-axis.


7.1 Volumes


Method of WashersEXAMPLE: Let D be the region of the xy-plane that is bounded above by and below by y = x2. Calculate the volume of the solid of revolution that is generated when D is rotated about the x-axis.


7.1 Volumes


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 Vof the solid obtained by rotating R about the x-axis is given by


7.1 Volumes


Method of Washers

EXAMPLE: Let R be the region of the xy-plane that is bounded above by y = ex, 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.


7.1 Volumes


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 .


7.1 Volumes


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.


7.1 Volumes


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.


7.1 Volumes


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


7.1 Volumes



EXAMPLE: Calculate the volume generated when the region bounded by y = x3 + x2, the x-interval [0, 1], and the vertical line x = 1 is rotated about the y-axis.


7.1 Volumes



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 generatedwhen the region bounded by x = g (y), the y-axis, y = c, and y = d is rotated about the x-axis is


7.1 Volumes



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 = y2 and x = y is rotated about the line y = 2.


7.1 Volumes


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 tocalculate the volume of the resulting solid of revolution, what is the variable of integration?

Chapter 7-Application of the Integral7.2 Arc Length and Surface

Area


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


7.2 Arc Length and Surface Area


Some Examples of Arc Length

EXAMPLE: Calculate the arc length L of the graph of f(x) = 2x3/2 over the interval [0, 7].

EXAMPLE: Calculate the length L of the graph of the function f(x) = (ex + e−x)/2 over the interval [1, ln (8)].




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 9x2 = 4y3 between the points (0, 0) and (2/3, 1).




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




Parametric Curves

EXAMPLE: Calculate the length L of the parametric curve C defined by the parametric equations x = 1 + 2t3/2 and y = 3 + 2t for 0 ≤ t ≤ 5.




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




Surface Area

EXAMPLE: Show that S = 4r2 is the surface area of a sphere of radius r.




Quick Quiz1. The arc length of the graph of y = x3 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 = et, y = t2 between (1, 0) and (e, 1) is equal to for what function g (t)?

3. The graph of y = x3 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?

Chapter 7-Application of the Integral7.3 The Average Value of a

Function


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


7.3 The Average Value of a Function


The Basic Technique

EXAMPLE: A rod of length 9 cm has temperature distribution T (x) =(2x − 6x1/2)C for 0 ≤ x ≤ 9. This means that, at position x on the rod, the temperature is (2x − 6x1/2 )C Calculate the average temperature of the rod.

EXAMPLE: What is the average value favg of the function f(x) = x2 on the interval [3, 6]?




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 fX is used to emphasize the association between f and X.




Random Variables

EXAMPLE: In a large class, the grades on a particular exam are all between 38 and 98. Let X denote thescore of a randomly selected student in the class. Suppose that the probability density function f for X is given by f (x) =(136x − 3724 − x2)/36000, 38 ≤ x ≤ 98. What is the probability that the grade on a randomly selected exam is between 72 and 82?




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.




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) = 20x3 (1 − x) for 0 ≤ x ≤ 1. What is average of X?




Population Density Functions

EXAMPLE: In 1950 the population density of Tulsa was given by f (x) = 28000 e−4x/5, where x represents thedistance in miles from the central business district. About how many people lived within 20 miles of the city center?




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) = x2 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?

Chapter 7-Application of the Integral7.4 Center of Mass


Moments

DEFINITION: Let c be any real number. Suppose that f is continuous and nonnegative on the interval [a, b]. Let Rdenote the planar region bounded above by the graph of y = f (x), below by the x-axis, and on the sides bythe line segments x = a and x = b. If R has a uniform mass density , then the moment Mx=c of R about theaxis x = c is defined by



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.



Center of MassDEFINITION: Let R be a region as shown below. Then the center of mass of R is the point (x, y) whosecoordinates satisfy



Center of Mass

THEOREM: Let f be a continuous nonnegative function on the interval [a, b]. Let R denote the region bounded aboveby 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 denotethe 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



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.



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 Rdenote 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



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 unitmass density. What is the center of mass of R?



Quick Quiz1 Let R denote the triangle with vertices (0, 0), (2, 0), and (2, 6). The x-center of mass of R is given by theequation

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 theequation

for what function g (x)?

Chapter 7-Application of the Integral7.5 Work


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


7.5 Work


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?


7.5 Work


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?


7.5 Work


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?


7.5 Work


Quick Quiz1. To overcome friction, a force of 12 − 3x2N 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?

Chapter 7-Application of the Integral7.6 First Order Differential

Equations-Separable Equations


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.


7.6 First Order Differential Equations-Separable Equations


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.




Slope Fields

EXAMPLE: Sketch a slope field for the differential equation, dy/dx= x − y.




Initial Value Problem

DEFINITION: If x0 and y0 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(x0) = y0 and y’(x) = F (x, y (x)) for all x in some open interval containing x0.




Initial Value Problem

EXAMPLE: Solve the initial value problem dy/dx = x − y, y (0) = 2.




Separable Equations

EXAMPLE: Solve the initial value problem




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




Examples from the Physical SciencesEXAMPLE: According to Torricelli’s Law, the rate at which water flows out of a tank from a small hole in thebottom 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 0.4m 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 m1/2/s, findthe height y (t) of water as a function of time. How long will it take for the tank to empty?




Quick Quiz1. 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(x3) and y()=2, then for what , and C is

Chapter 7-Application of the Integral7.7 First Order Differential

Equations-Linear Equations


Solving Linear Differential Equations

THEOREM: Suppose that p (x) and q (x) are continuous functions. Let P (x) be any antiderivative of p (x). Thenthe general solution of the linear equation

is

where C is an arbitrary constant.


7.7 First Order Differential Equations-Linear Equations


Solving Linear Differential Equations

EXAMPLE: Solve the initial value problem

EXAMPLE: Find the general solution of the linear differential equation




An Application: Mixing Problems

EXAMPLE: A 200 gallon tank is filled with a salt solution initially containing 10 pounds of salt. An inlet pipebrings a solution of salt in at the rate of 10 gallons per minute. The concentration of salt in the incoming solution is1−e−t/60 pounds per gallon when t is measured in minutes. An outlet pipe prevents overflow by allowing 10 gallonsper minute to flow out of the tank. How many pounds of salt are in the tank at time t? Long term, about how manypounds of salt will be in the tank?




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




Newton’s Law for Temperature Change

THEOREM: Suppose that T0 = T (0) is the temperature of an object when it is placed in an environment that hasconstant temperature T. Suppose that the temperature T (t) of the object is governed by Newton’s law of temperaturechange. That is, suppose that T (t) is a solution of equation

then




Newton’s Law for Temperature Change

EXAMPLE: A thermometer is at room temperature (20.0C). One minute after being placed in a patient’s throat it reads 38.0C. One minute later it reads 38.3C. Is this second reading an accurate measure (to three significant digits) of the patient’s temperature?




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 ?

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