MCS 122 Chapter 5
Review of Basic Integration
Some of the material in these slides is from Calculus 9/E by
Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Antiderivatives
Definition 5.2.1 (p. 322)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Table 5.2.1 (p. 324)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Theorem 5.2.3 (p. 325)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Equations (4) – (7) (p. 326)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Antiderivatives = Family of
Functions
1 3
 x dx  3 x  C
2
Figure 5.2.1 (p. 327)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
 7
2 
1. Evaluate    sec x dx
5x

a) 7 x 2  tan x  C
10


b) 7 ln x  tan x  C
5

c) 7 ln x  tan x  C
5

d) 7 ln x  1 sec 3 x  C
5
3
Question 1
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
The Area Problem
5.1.1 (p. 317)
The Area Problem
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Figure 5.1.4 (p. 318)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Rectangular
Approximation for
n
Area =
*
 f (x
k 1
k
) x
This is called a
“Riemann Sum” for
the area.
Figure 5.4.4 (p. 344)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definition of Area for a
continuous function
Definition 5.4.3 (p. 344)
Area Under a Curve
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Figure 5.1.4 (p. 318)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Riemann sums yield “signed” area
Figure 5.4.10 (p. 349)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Three Common Approximations
Figure 5.4.7 (p. 347)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Approximations with variable
width
n
 f (x
k 1
*
k
) x k
Figure 5.5.1 (p. 353)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral
Definition 5.5.1 (p. 354)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Continuous Functions are
Integrable
Theorem 5.5.2 (p. 355)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral Rules
Definition 5.5.3 (p. 356)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral Rules
Theorem 5.5.4 (p. 357)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral Rules
Theorem 5.5.5 (p. 358)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral vs Antiderivatives
We have seen two basic ideas so far:
Antiderivative: Computes a family of functions

f ( x)dx
Definite Integral: Computes a number = area

b
a
f ( x)dx
Is there a connection between these two?
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Fundamental Theorem of
Calculus - I
Theorem 5.6.1 (p. 363)
The Fundamental Theorem of Calculus, Part 1
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Fundamental Theorem of
Calculus - I
Practice
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Exercise 5.6.69 (p. 375)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Fundamental Theorem of
Calculus - II
Theorem 5.6.3 (p. 370)
The Fundamental Theorem of Calculus, Part 2
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
U-Substitution
Guidelines for u-Substitution (p. 334)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
2. Evaluate 
8
1
 2

2
5x 3  4 x dx


a) 89.5

b) 97.5
c) 96.5
Question 2
d) -89.5
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
3. Find the area under the curve y  x over the interval [9, 16].
a) 91

b) 24 2
3
c) 37

d) 24 2
3
Question 3
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.


Select the best answer for the following question.
4. Use part 2 of the Fundamental Theorem of Calculus to find:

d  x
t3
dt
1
dx  tant  1 
2
3x
a)
sec 2 x 1
t 3 
b)
tant 1
2
3
2
c) 3x tanx 1  x sec x 1
tan 2 x 1
d)
3
x
tanx 1
Question 4
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.



Select the best answer for the following question.
6. Evaluate
 2x  3
7
dx.
a) 1 2x  3 8  C

 
8
b) 142x  36  C
c) 1 2x  38  C
16
d) 1 2x  38  C
2
Question 6
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Answers
1. c
2. a
3. b
4. d
5.
6. a
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.