chapter06

Chapter 6. Integration

• Overview of Area Problem • The Indefinite Integral • Integration by Substitution • The Definition of Area as a Limit; Sigma Notation • The Definite Integral • Rectilinear Motion Revisited Using Integration • Evaluation Definite Integrals by Substitution • Logarithmic Functions from the Integral Point of View Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Method of finding areas Rectangle method 5

A

5 

i

 1

f x



x n A n



i

 1

f x



x n

lim

A n



A

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Antiderivative method If f is a nonnegative continuous function on the interval [a, b], and if A(x) denotes the area under the graph of f over the interval [a, x], where x is any point in the interval [a, b], then

A′(x) = f(x)

Ex. 2.1-2 (page 353) Use the antiderivative method to find the area under the graph of y=x

2

over the interval [0, 1].



x

2  1 3

x

3 

C

Area over the interval [0, 0] = 0  1 3

x

3 

A

(0) 0 

A

(1)  1 3 Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

6.2.2. THEOREM

.

If F(x) is any antiderivative of f(x) on an interval I, then for any constnat C the function F(x) + C is also an antiderivative on that interval. Moreover, each antiderivative of f(x) on the interval I can be expressed in the form F(x) + C by choosing the constant C appropriately.

( ) 

C

The integration of f(x) with respect to x is equal to F(x) plus a constant Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Integration formulas Q. 6.2.35 Evaluate the integral 1

x dx

§Hint. Multiply the numerator and denominator by an appropriate expression Multiply the integrand by 1 sin 1 sin

x x

� 2

x x dx

 2

x x

 2

x

  tan

x

 sec

x


Properties of the Indefinite Integral Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Differential Equations

dy dx

 f(x) is a known function and y=F(x), where F(x) is an unknown function Initial value problem

dy dx

 

y

0 Q. 6. 2. 49 At each point (x, y) on the curve, y satisfies the condition d

2 y/dx 2

=6x; the lin y=5-3x is tangent to the curve at the point where x=1.

dy dx

 3

x

2 

C

1

dy dx x

 1  2 

C

1

C

1 6

y x

6 2

y x

 1

y x



C

2 

C

2  7 6

x

 7 Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Integration by Substitution

d dx

   ( ( )) 

C

( ( )) 

C

( ) 

C

Ex. 6.3.4

� 1 �

dx x

 8  �

du

5  3 �   3 4

u

 4 '( ) 3 1 �

x

 8 � � 4 

C


Guidelines for u-Substitution


Equation (5) (p. 370) Equation (6) Equation (7)


Theorem 6.4.1 (p. 375)


Theorem 6.4.2 (p. 376)


Figure 6.4.4 (p. 377)


Definition 6.4.3 (p. 378)

Area Under a Curve Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 6.4.7 (p. 380)


Definition 6.4.5 (p. 381)

Net Signed Area Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 6.4.9 (p. 382)


Figure 6.5.1 (p. 386)


Definition 6.5.1 (p. 387)


Theorem 6.5.2 (p. 388)


Definition 6.5.3 (p. 390)


Theorem 6.5.4 (p. 391)


Theorem 6.5.5 (p. 391) Theorem 6.5.6 (p. 392)


Theorem 6.6.1 (p. 397)

The Fundamental Theorem of Calculus, Part 1 Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 6.6.4 (p. 400)


Figure 6.6.8 (p. 402)


Figure 6.6.9 (p. 405)


Theorem 6.6.2 (p. 402)

The Mean-Value Theorem for Integrals Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 6.6.3 (p. 403)

The Fundamental Theorem of Calculus, Part 2 Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

6.6.4 (p. 405)

Integrating a Rate of Change Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Exercise 6.6.73 (p. 409)


Equation (3) (p. 411) Equation (4)


Figure 6.7.3 (p. 412)


Exercise 6.7.3 (p. 416)


Theorem 6.8.1 (p. 420)


Definition 6.9.1 (p. 425)


Figure 6.9.2 (p. 426)


Theorem 6.9.2 (p. 426)


Theorem 6.9.3 (p. 428)


Definition 6.9.4 (p. 429) Theorem 6.9.5 (p. 629)


Definition 6.9.6 (p. 430) Definition 6.9.9 (p. 431)


Exercise 6.9.27 (p. 435)


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Chapter 6. Integration

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