### Main Menu

Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

### Section 5.1 An Area Problem; A Speed-Distance Problem

### a. An Area Problem

### b. An Area Problem (continued)

### c. Upper Sums and Lower Sums

### d. Overview of the Speed Distance Problem

### Section 5.2 The Definite Integral of a Continuous Function

### a. Partitions

### b. Partitions; Upper Sum and Lower Sum

### c. Example

### d. Definition 5.2.3

### e. Use of "dummy" Variables

### f. Integral of a Constant Function

### g. Integral of the Identity Function

### h. The Integral as the Limit of Riemann Sums

### Section 5.3 The Function

### a. Theorem 5.3.1

### b. The Effects of Adding Points to a Partition

### c. Theorem 5.3.2

### d. Properties of Integration

### e. Theorem 5.3.5

### f. *Integration when f > 0*

### g. Example

### Section 5.4 The Fundamental Theorem of Integral Calculus

### a. Antiderivatives and the Fundamental Theorem of Integral Calculus

### b. Example

### c. Some Common Antiderivatives and Examples

### d. Linearity of the Integral

### e. Example

### Chapter 5: Integration

### Section 5.5 Some Area Problems

### a. Area of Ω

### b. Signed Area

### Section 5.6 Indefinite Integrals

### a. Indefinite integrals

### b. Common Indefinite Integrals

### c. Linearity Properties

### d. Application to Motion Example

### Section 5.7 Working Back from the Chain Rule; The u-Substitution

### a. Theorem 5.7.1

### b. Example

### c. Substitution in Definite Integrals

### Section 5.8 Additional Properties of the Definite Integral

### a. Properties I and II

### b. Properties III and IV

### c. Property V

### d. Property VI

### Section 5.9 Mean-Value Theorems for Integrals; Average Value of a Function

### a. First Mean-Value Theorem for Integrals

### b. Area of Ω

### c. Second Mean-Value Theorem for Integrals

### ( )

^{x}### ( )

*a*

*F x* = ∫ *f t dt*

### An Area Problem; A Speed-Distance Problem

### In Figure 5.1.1 you can see a region Ω bounded above by the graph of a continuous

*function f, bounded below by the x-axis, bounded on the left by the line x = a, and bounded * *on the right by the line x = b. The question before us is this: What number, if any, should be * called the area of Ω?

### To begin to answer this question, we split up the interval *[a, b] into a finite number of subintervals*

*[x* _{0} *, x* _{1} *], [x* _{1} *, x* _{2} *], . . . , [x* _{n−1} *, x* _{n} *] with a = x* _{0} *< x* _{1} *< · · · < x* _{n} *= b* *.* This breaks up the region *Ω into n subregions:*

_{n−1}

_{n}

_{n}

### Ω 1 *, * Ω 2 *, . . . , * Ω *n* *. * (Figure 5.1.2)

### We can estimate the total area of Ω by estimating the area

### of each subregion Ω _{i} and adding up the results.

_{i}

### Main Menu

Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

### An Area Problem; A Speed-Distance Problem

### Adding up these inequalities, we get on the one hand

### and on the other hand

### An Area Problem; A Speed-Distance Problem

### A sum of the form

*m* _{1} *Δx* _{1} *+ m* _{2} *Δx* _{2} *+· · ·+m* _{n} *Δx* _{n} (Figure 5.1.4)

_{n}

_{n}

*is called a lower sum for f.*

### A sum of the form

*M* _{1} *Δx* _{1} *+ M* _{2 } *Δx* _{2} *+· · ·+ M* _{n } *Δx* _{n} (Figure 5.1.5)

_{n }

_{n}

*is called an upper sum for f.*

### For a number to be a candidate for the title “area of Ω,” it must be greater than or

### Main Menu

Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

### An Area Problem; A Speed-Distance Problem

### If an object moves at a constant speed for a given period of time, then the total distance traveled is given by the familiar formula

*distance = speed × time.*

### Suppose now that during the course of the motion the speed ν does not remain constant;

### suppose that it varies continuously. How can we calculate the distance traveled in that case?

*To answer this question, we suppose that the motion begins at time a, ends at time* *b, and during the time interval [a, b] the speed varies continuously.*

*As in the case of the area problem, we begin by breaking up the interval [a, b] into* a finite number of subintervals:

*[t* _{0} *, t* _{1} *], [t* _{1} *, t* _{2} *], . . . , [t* _{n} _{−1} *, t* _{n} *] with a = t* _{0} *< t* _{1} *< · · · < t* _{n} *= b.*

_{n}

_{n}

_{n}

*On each subinterval [t* _{i} _{−1} *, t* _{i} *] the object attains a certain maximum speed M* _{i} and a *certain minimum speed m* _{i} .

_{i}

_{i}

_{i}

_{i}

*The total distance traveled during the full time interval [a, b], call it s, must be the sum of * *the distances traveled during the subintervals [t* _{i} _{−1} *, t* _{i} ]; thus we must have

_{i}

_{i}

*s = s* _{1} *+ s* _{2} *+ · · · + s* _{n} *.*

_{n}

*Similar to the area problem it can be shown that s must be greater than or equal to every *

### lower sum for the speed function, and it must be less than or equal to every upper sum. It

### turns out that there is one and only one such number, and this is the total distance traveled.

### The Definite Integral of a Continuous Function

**Example** The sets

*{0, 1}, {0, ½, 1}, {0, ¼, ½, 1}, {0,* ^{1} / 4 *, * ^{1} / 3 *, * ^{1} / 2 *, * ^{5} / 8 *, 1}*

### are all partitions of the interval [0, 1].

### Main Menu

### The Definite Integral of a Continuous Function

*If P = {x* _{0} *, x* _{1} *, x* _{2} *, . . . , x* _{n−1} *, x* _{n} *} is a partition of [a, b], then P breaks up [a, b]*

_{n−1}

_{n}

*into n subintervals*

*[x* _{0} *, x* _{1} *], [x* _{1} *, x* _{2} *], . . . , [x* _{n−1} *, x* _{n} ] of lengths *Δx* _{1} *, * *Δx* _{2} *, . . . , * *Δx* _{n} *.*

_{n−1}

_{n}

_{n}

*Suppose now that f is continuous on [a, b]. Then on each subinterval [x* _{i−1} *, x* _{i} ]

_{i−1}

_{i}

*the function f takes on a maximum value, M* _{i} *, and a minimum value, m* _{i} .

_{i}

_{i}

### The Definite Integral of a Continuous Function

**Example **

*The function f (x) = 1 + x* ^{2} *is continuous on [0, 1]. The partition P = {0, ½, ¾, 1} *

*breaks up [0, 1] into three subintervals*

*[x* _{0} *, x* _{1} *] = [0, ½], [x* _{1} *, x* _{2} ] = [ ½ *, * ¾] *, [x* _{2} *, x* _{3} ] = [ ¾ *, 1] *

### of lengths

*Δx* 1 = ½ _{− 0 = } ½ *, * Δ ^{x} 2 = ¾ _{− } ½ *= ¼, * Δ ^{x} 3 = 1 − ¾ = ¼ *.* *Since f increases on [0, 1], it takes on its maximum value at the right * endpoint of each subinterval:

^{x}

^{x}

### The minimum values are taken on at the left endpoints:

### Thus

### ( )

### 1 2 3

### 1 5 3 25

### , , 1 2

### 2 4 4 16

*M* = *f* = *M* = *f* = *M* = *f* =

### ( )

### 1 2 3

### 1 5 3 25

### 0 1, ,

### 2 4 4 16

*m* = *f* = *m* = *f* = *m* = *f* =

### ( ) ^{5 1} ^{25 1} ^{2} ^{1} ^{97} ^{1.52}

*U* *P* = *M* ∆ + *x* *M* ∆ + *x* *M* ∆ = *x* + + = ≅

### Main Menu

### The Definite Integral of a Continuous Function

### The Definite Integral of a Continuous Function

### In the expression

*the letter x is a “dummy variable”; in other words, it can be replaced by any letter * not already in use. Thus, for example,

*all denote exactly the same quantity, the definite integral of f from a to b.*

*From the introduction to this chapter, you know that if f is nonnegative and *

*continuous on [a, b], then the integral of f from x = a to x = b gives the area below * *the graph of f from x = a to x = b:*

### ( )

*b*

*a* *f x dx*

### ∫

### ( ) ( ) ^{,}

*b*

*a* *f x d x*

### ∫ ∫ *a* ^{b} *f t dt* ^{( )} ^{,} ∫ *a* ^{b} *f z dz* ^{( )}

^{b}

^{b}

### ( ) ^{.}

*A* = ∫ *b* *f x dx*

### Main Menu

### The Definite Integral of a Continuous Function

### The integral of a constant function as shown in Figure 5.2.3:

### The Definite Integral of a Continuous Function

### The integral of the identity function as shown in Figure 5.2.5:

### Main Menu

### The Integral as the Limit of Riemann Sums

### Figure 5.2.8 illustrates the idea that the definite integral of a continuous function is the limit of Riemann sums . Here the base interval is broken up into eight

### subintervals. The point *is chosen from [x* _{0} *, x* _{1} ], *from [x* _{1} *, x* _{2} ], and so on.

### which in expanded form reads

### *

*x* 1 *x* _{2} ^{*}

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

### Main Menu

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

### Main Menu

*Until now we have integrated only from left to right: from a number a to a * *number b greater than a. We integrate in the other direction by defining*

### The integral from any number to itself is defined to be zero:

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

### Main Menu

*F(x) = area from a to x and F(x + h) = area from a to x + h. Therefore* *F(x + h) – F(x) = area from x to x + h. For small h this is approximately * *f (x) h. Thus*

### ( ) ( )

*F x* *h* *F x* *h*

### + − is approximately ^{f x h} ( ) ^{f x} ( )

^{f x h}

^{f x}

*h* =

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

**Example**

### Set *for all real numbers x.*

**(a) Find the critical points of F and determine the intervals on which F increases and the ** *intervals on which F decreases.*

**(a) Find the critical points of F and determine the intervals on which F increases and the**

**(b) Determine the concavity of the graph of F and find the points of inflection (if any).**

**(b) Determine the concavity of the graph of F and find the points of inflection (if any).**

**(c) Sketch the graph of F.**

**(c) Sketch the graph of F.**

**Solution**

**(a) To find the intervals on which F increases and the intervals on which F decreases,** *we examine the first derivative of F. By Theorem 5.3.5,*

**(a) To find the intervals on which F increases and the intervals on which F decreases,**

*for all real x.*

*Since F* ´ *(x) > 0 for all real x, F * *increases on (−∞,∞); there are no critical points.*

**(b) To determine the concavity of the graph and to find the points of inflection, we use** the second derivative

*The sign of F* ´´ *and the behavior of the graph of F are as follows:*

### ( ) _{2}

### 0

### 1 1

*F x* *x* *dt*

### = *t*

### ∫ +

### ( ) ^{1} _{2}

*F x* 1

### ′ = *x* +

### ( ) ( ^{1} ^{2} ^{x} ^{2} ) ^{2}

^{x}

*F* *x*

*x*

### ′′ = − +

### ( )

### The Function ( )

*x*

*a*

*F x* = ∫ *f t dt*

### Main Menu

### The Fundamental Theorem of Integral Calculus

### The Fundamental Theorem of Integral Calculus

**Example** Evaluate

**Solution**

*As an antiderivative for f (x) = x* ^{2} , we can use the function *G(x) = * *⅓x* ^{3} *. *

### By the fundamental theorem,

*NOTE: Any other antiderivative of f (x) = x* ^{2} *has the form H(x) = * *⅓x* ^{3} *+ C for* *some constant C. Had we chosen such an H instead of G, then we would * have had

### 4 2

### 1 *x dx*

### ∫

### ( ) ( ) ( ) ( )

### 4 2 3 3

### 1

### 1 1 64 1

### 4 1 4 1 21

### 3 3 3 3

*x dx* = *G* − *G* = − = − =

### ∫

###

### Main Menu

### The Fundamental Theorem of Integral Calculus

### Some examples:

### The Linearity of the Integral

**I. Constants may be factored through the integral sign:**

**II. The integral of a sum is the sum of the integrals:**

**III. The integral of a linear combination is the linear combination of the integrals:**

### Main Menu

### The Linearity of the Integral

**Example** Evaluate

### [ ]

### 4

### 0 ^{π} sec *x* 2 tan *x* − 5sec *x dx*

### ∫

**Solution**

### [ ]

### [ ] [ ]

### [ ]

### 4 4

### 2

### 0 0

### 4 4

### 2

### 0 0

### 4 4

### 0 0

### sec 2 tan 5sec 2sec tan 5sec

### 2 sec tan 5 sec

### 2 sec 5 tan

### 2 sec sec 0 5 tan tan 0

### 4 4

### 2 2 1 5 1 0 2 2 7

*x* *x* *x dx* *x* *x* *x dx*

*x* *x dx* *x dx*

*x* *x*

### π π

### π π

### π π

### π π

###

### − = −

### = −

### = −

###

### = − − −

###

### = − = − = −

### ∫ ∫

### ∫ ∫

### Some Area Problems

### Look at the region Ω shown in Figure 5.5.4. The upper boundary of Ω is the graph of a *nonnegative function f and the lower boundary is the graph of a nonnegative function g. *

### We can obtain the area of Ω by calculating the area of Ω _{1} and subtracting off the area of Ω _{2} . Since

### ( ) ( )

### 1 2

### area of ^{b} and area of ^{b}

^{b}

^{b}

*a* *f x dx* *a* *g x dx*

### Ω = ∫ Ω = ∫

### we have

### Main Menu

### Some Area Problems

*The area between the graph of f and the x-axis from x = a to x = e is the sum * area of Ω _{1} + area of Ω _{2} + area of Ω _{3} + area of Ω _{4}

### This area is

### ( ) ( ) ( ) ( ) ^{.}

*b* *c* *d* *e*

*a* *f x dx* − *b* *f x dx* + *c* *f x dx* − *d* *f x dx*

### ∫ ∫ ∫ ∫

### Indefinite Integrals

*Consider a continuous function f . If F is an antiderivative for f on [a, b], then*

*If C is a constant, then*

### Thus we can replace (1) by writing

*If we have no particular interest in the interval [a, b] but wish instead to emphasize that* *F is an antiderivative for f , which on open intervals simply means that F* ´ *= f , then* *we omit the a and the b and simply write*

### ( ) ( )

### (1) ^{b} ^{b}

^{b}

^{b}

*a* *f x dx* = *F x* *a*

### ∫

### ( ) ^{b} _{a} ( ) ( ) ( ) ( ) ( ) ^{b} _{a}

^{b}

_{a}

^{b}

_{a}

*F x* *C* *F b* *C* *F a* *C* *F b* *F a* *F x*

### + = + − + = − =

###

### ( ) ( ) ^{.}

*b* *b*

*a* *f x dx* = *F x* + *C* *a*

### ∫

### ( ) ( )

*f x dx* = *F x* + *C*

### ∫

### Main Menu

### Indefinite Integrals

### Indefinite Integrals

### The linearity properties of definite integrals also hold for indefinite integrals.

**Example**

### Calculate ∫ 5 *x* ^{3 / 2} − 2 csc ^{2} *x dx* **Solution**

### ( )

### 3 / 2 2 3 / 2 2

### 5 / 2

### 1 2

### 5 / 2

### 5 2 csc 5 2 csc

### 5 2 2 cot

### 5

### 2 2 cot

*x* *x dx* *x* *dx* *xdx*

*x* *C* *x* *C*

*x* *x* *C*

### − = −

###

### = + − − +

###

### = + +

### ∫ ∫ ∫

*(writing C for C* *+ C* )

### Main Menu

### Indefinite Integrals

**Application to Motion** **Example**

### An object moves along a coordinate line with velocity *v(t) = 2 − 3t + t* ^{2} *units per second.*

*Its initial position (position at time t = 0) is 2 units to the right of the origin. Find * the position of the object 4 seconds later.

**Solution**

*Let x(t) be the position (coordinate) of the object at time t. We are given* *that x(0) = 2. Since x* ´ *(t) = v(t),*

*Since x(0) = 2 and x(0) = 2(0) − * ^{3} ∕ ^{2} (0) ^{2} + ⅓(0) ^{3} *+ C = C, we have C = 2 and* *x(t) = 2t − * ^{3} ∕ ^{2} ^{t} ^{2} + ⅓t ^{3} *+ 2.*

^{t}

*The position of the object at time t = 4 is the value of this function at t = 4:*

*x* (4) = 2(4) − ^{3} ∕ ^{2} ^{(4)} ^{2} ^{+ } ⅓(4) ^{3} + 2 = 7 ⅓

### At the end of 4 seconds the object is 7 ⅓ units to the right of the origin.

### The motion of the object is represented schematically in Figure 5.6.1.

### ( ) ( ) ( ^{2 3} ^{2} ) ^{2} ^{3} ^{2} ^{1} ^{3}

### 2 3

*x t* = ∫ *v t dt* = ∫ − + *t* *t* *dt* = *t* − *t* + *t* + *C*

*The u-Substitution*

### Main Menu

*The u-Substitution*

**Example** Calculate **Solution**

*Set u = 3 + 5x, du = 5 dx. Then*

### ( ) ^{2}

### 1 3 5

*dx* + *x*

### ∫

### ( )

### ( ) ( )

### 2

### 2 2

### 2 1

### 2

### 1 1 1 1

### 5 5

### 3 5

### 1 1 1 1

### 5 5 5 3 5

### 3 5

*dx* *du* *u du*

*x* *u*

*dx* *u du* *u* *C* *C*

*x* *x*

### −

### − −

###

### = = +

### = = − + = − +

### + +

### ∫ ∫

### and

*The u-Substitution*

**The Definite Integral **

*This formula is called the change-of-variables formula. The formula can be * *used to evaluate provided that u* ´ *is continuous on [a, b] *

*and f is continuous on the set of values taken on by u on [a, b]. Since u is * *continuous, this set is an interval that contains a and b.*

### ( ( ) ) ^{( )}

*b*

*a* *f u x u x dx* ′

### ∫

### Main Menu

### Additional Properties of the Definite Integral

**I. The integral of a nonnegative continuous function is nonnegative:**

### The integral of a positive continuous function is positive:

**II. The integral is order-preserving: for continuous functions f and g,**

**II. The integral is order-preserving: for continuous functions f and g,**

### and

### Additional Properties of the Definite Integral

**III. Just as the absolute value of a sum of numbers is less than or equal to the sum ** of the absolute values of those numbers,

*|x* _{1} *+ x* _{2} *+· · ·+ x* _{n} *| ≤ |x* _{1} *| + |x* _{2} *|+· · ·+|x* _{n} *|,*

_{n}

_{n}

### the absolute value of an integral of a continuous function is less than or equal to the integral of the absolute value of that function:

**IV. If f is continuous on [a, b], then**

**IV. If f is continuous on [a, b], then**

### Main Menu

### Additional Properties of the Definite Integral

**V. If f is continuous on [a, b] and u is a differentiable function of x with ** *values in [a, b], then for all u(x) ∈ (a, b)*

**V. If f is continuous on [a, b] and u is a differentiable function of x with**

**Example** Find

**Solution**

### At this stage you probably cannot carry out the integration: it requires the

### natural logarithm function. (Not introduced in this text until Chapter 7.) But for our purposes, that doesn’t matter. By (5.8.7),

3

### 0

### 1 1 *d* *x*

*dx* *t* *dt*

###

### +

### ∫

### 3 2

### 2

### 3 3

### 0

### 1 1 3

### 1 1 3 1

*d* *x* *x*

*dt* *x*

*dx* *t* *x* *x*

### = =

### + + +

### ∫

### Additional Properties

**VI. Now a few words about the role of symmetry in integration. Suppose that f is**

**VI. Now a few words about the role of symmetry in integration. Suppose that f is**