Download Calculus

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: A function F is called an antiderivative of a
function f on an interval I if F’(x) = f(x) for all x in I.
Definition: The notation  f (x) dx is used for an
antiderivative of f and is called an indefinite integral.

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.
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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Integration Practice – Solutions at end of Slides
 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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
To compute area we use
the idea of approximating
rectangles.
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 =

k 1
f ( x k* ) x
This is called a
“Riemann Sum” for
the area.
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
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
Riemann Sum area
= area above x-axis
– area below x-axis
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Three Common Approximations
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral
Definition of Definite Integral: If f(x) is a function defined for
a ≤ x ≤ b, we divide the inetrval [a,b] into n subintervals of equal
width Δx = (b-a)/n. We let x0 =a, x1 , x2 , …, xn (=b) be the
endpoints of these subintervals and we let x1* , x2* , …, xn* be
any sample points in these subintervals. Then, the definite
integral of f from a to b is
b

a
n
f (x) dx  lim  f (x *i ) x
n
i1
provided this limit exists and gives the same value for all possible
choices of sample points. If the limit exists, we say f is integrable
on [a,b]

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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral Rules
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral Rules
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definite Integral Rules
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 – Part I
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 – Part II
(Also called the Net Change Theorem in Stewart)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Integration Practice – Solutions at end of Slides
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.

Integration Practice – Solutions at end of Slides
3. Find the area under the curve y  x over the interval [9, 16].
a) 91

2
b) 24
3
c) 37
2
d) 24
3
Question 3
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.


Integration Practice – Solutions at end of Slides
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.
Answers
1. c
2. a
3. b
4. d
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.